academic intervention: acceleration and …
TRANSCRIPT
ACADEMIC INTERVENTION: ACCELERATION AND REMEDIATION
by
Barbara Gail Franklin
Liberty University
A Dissertation Presented in Partial Fulfillment
Of the Requirements for the Degree
Doctor of Education
Liberty University
2016
2
ACADEMIC INTERVENTION: ACCELERATION AND REMEDIATION
by Barbara Gail Franklin
A Dissertation Presented in Partial Fulfillment
Of the Requirements for the Degree
Doctor of Education
Liberty University, Lynchburg, VA
2016
APPROVED BY:
Kenneth Gossett, Ph.D., Committee Chair
Ralph Marino, Jr., Ed.D. Committee Member
Jason Palmer, Ph.D., Committee Member
Scott Watson, Ph.D., Associate Dean, Advanced Programs
3
ABSTRACT
Eighth grade math students must pass a standards based test to be promoted to the next grade.
Students who were at risk of failing the state’s annual test faced impending retention. The
purpose of this quasi-experimental study was to see if an intensive nine-week (55 min per day)
remedial Math Connection (MC) class for 67 suburban, eighth grade students identified as at risk
of failing, could significantly increase the scores; concurrently, at this Title I school, they were
compared with 122 eighth grade students who were not identified as at risk of failing. The
dependent variable was measured using the AIMSweb tests (nonmultiple choice answer format).
A quantitative quasi-experiment of nonequivalent control group design, pretest and posttest, was
used with the AIMSweb tests. When controlling for pretest scores through an analysis of
covariance (ANCOVA), results indicated that there was no significant difference between the
AIMSweb scores for the math class group as compared with the no math class group. Future
studies need to consider both efficient and effective processes of instruction and assessment
formats for the remediation of students at risk of failing the state’s math summative assessment.
Keywords: math, remediation, FAPE, assessment, time on task, middle school
4
Dedication
I would like to dedicate this work to my husband David, to my children and their spouses,
and my grandchildren. I have felt loved, encouraged, and supported throughout the process of
obtaining my Doctorate of Education.
5
Acknowledgments
I would like to acknowledge my husband, David, who went above and beyond my
expectations in helping me to achieve my goal. In addition, I would like to acknowledge
my very patient and kind dissertation chair, Dr. Kenneth Gossett, who was an invaluable
asset in assisting me through this process. My thanks also extend to the teachers who
supported my efforts and their insights on the best ways in which to educate our nation’s
children.
6
Table of Contents
ABSTRACT .....................................................................................................................................3
Dedication ................................................................................................................4
Acknowledgments....................................................................................................5
List of Tables ...........................................................................................................8
List of Figures ..........................................................................................................9
List of Abbreviation ...............................................................................................10
CHAPTER ONE: INTRODUCTION ............................................................................................11
Background ............................................................................................................11
Problem Statement .................................................................................................15
Purpose Statement ..................................................................................................17
Significance of the Study .......................................................................................17
Research Question .................................................................................................19
Null Hypothesis .....................................................................................................19
Definitions..............................................................................................................19
CHAPTER TWO: LITERATURE REVIEW ................................................................................21
Introduction ............................................................................................................21
Theoretical Framework ..........................................................................................24
Related Literature...................................................................................................27
Formative Assessment ...........................................................................................30
Testing Format .......................................................................................................51
CHAPTER THREE: METHODS ..................................................................................................57
Research Design.....................................................................................................57
7
Research Question .................................................................................................57
Null Hypothesis .....................................................................................................58
Participants and Setting..........................................................................................58
Instrumentation ......................................................................................................60
Procedures ..............................................................................................................62
Data Analysis .........................................................................................................63
CHAPTER FOUR: FINDINGS ....................................................................................................65
Research Question .................................................................................................65
Null Hypothesis .....................................................................................................65
Descriptive Statistics ..............................................................................................65
Results ....................................................................................................................66
CHAPTER FIVE: DISCUSSION, CONCLUSIONS, AND RECOMMENDATIONS ..............73
Discussion ..............................................................................................................73
Conclusions ............................................................................................................76
Implications............................................................................................................79
Limitations .............................................................................................................80
Recommendations for Future Research .................................................................82
REFERENCES ..............................................................................................................................84
APPENDIX A SCHOOL APPROVAL .......................................................................................118
APPENDIX B IRB APPROVAL ................................................................................................120
8
List of Tables
Table 3.1: Frequences Demographics……………………………………………………………59
Table 4.1: Descriptive Statistics…....……………………………………………………………67
Table 4.2: Kolmogorov-Smirnov Test of Normality……………………………….…...............68
Table 4.3: Levene’s Test of Homogeneity of Variance….………….………………………...…69
Table 4.4: ANCOVA Table: Test of Homogeneity of Slopes………………………………….70
Table 4.5: ANCOVA Adjusted Mean Scores for the Treatment Groups……….…………….....71
Table 4.6: ANCOVA Table: Assessing Difference in AIMSweb Posttest Scores......……..……72
9
List of Figures
Figure 4.1: Box and whisker—AIMSweb for the control and treatment group ..……………….67
Figure 4.2: Scatterplots of AIMSweb scores for the control group…………….………………69
Figure 4.3: Scatterplot of AIMSweb scores for the treatment group..……………………...........70
10
List of Abbreviation
Common Core State Standards (CCSS)
Criterion Referenced Competency Test (CRCT)
Curriculum Based Measures (CBM)
Free Appropriate Public Education (FAPE)
General Linear Model (GLM)
Learning-Focused Schools (LFS)
Math Connection (MC)
Math Difficulty (MD)
No Child Left Behind (NCLB)
Response to Intervention (RTI)
Teacher Keys Effective System (TKES)
11
CHAPTER ONE: INTRODUCTION
Background
The Nation’s 2005 Report Card indicated that 31% of eighth grade math students lacked
proficiency (U.S. Department of Education, 2006). With the high stakes testing and the ending
of social promotion, students were at risk of retention based upon a state’s annual assessment
(Huddleston, 2014). In order for an eighth grade student in the state of Georgia to have access to
the subsequent high school math curriculum, the student must be able to meet the eighth grade
standards as measured by the state’s end-of-the-year testing instrument, the Criterion Referenced
Competency Test (CRCT). However, research has suggested that certain students have
significant gaps in math and that these inequities deny access to future math curriculum
(Dougherty, Goodman, Litke, & Page, 2015; Lukas & Beresford, 2010; Rickles, 2013; Rojas-
LeBouef & Slate, 2012). Bishop and Forgasz (2007) suggested “without access to mathematics
education there can be no equity” (p. 1146). In light of the inequities, educators have a two-fold
fiduciary responsibility—seek to identify those who are at risk of failing and implement a series
of interventions to bridge the academic gaps (Archer & Hughes, 2011; Courtade, Spooner,
Browder, & Jimenez, 2012; Petscher, Young-Suk, & Foorman, 2012).
Historical Literature Overview
The history of differentiated instruction has its roots within early America’s one room
school houses (Urban & Wagner, 2009). In this setting, one teacher was responsible for
educating students in a wide range of grades and ability levels. Some early American schools
used test based assessments to determine a student’s academic future (Huddleston, 2014; White,
1886; 1888). Some have estimated that in 1919 there were 190,000 one room schoolhouses in
the United States (Gundlach, 2012; Urban & Wagner, 2009). In early 1889, Preston Search, a
12
school superintendent in Colorado, advocated that teachers should make it possible for students
to work at their own pace without the fear of retention or failure (Urban & Wagner, 2009;
Ventura, 2014). Search pushed his teachers to build an environment where students could be
successful, progressing at the individual’s pace. However, by 1912, with the introduction of
assessment tests, significant academic gaps were identified (Urban & Wagner, 2009; Ventura,
2014).
These academic gaps, along with the implementation of intelligence tests, suggested
significant academic abilities existed between students. By the 1930s student ability and
readiness to learn a certain concept or skill would soon be eclipsed by a pedagogy suggesting
that students need to learn the way teachers teach and within the allotted time (Urban & Wagner,
2009; Ventura, 2014). A dichotomy of responses emerged. Students who did not learn were
retained or socially promoted; however, retention often resulted in an increase of students
"dropping out” (Allensworth, 2005; 2010; Jacob & Lefgren, 2009; Educational Commission of
the States, 2005; Xia & Kirby, 2009).
Society-at-large Discussion
In the shadow of students dropping out emerged Federal legislation based on equal access
and minimal outcomes. On the heels of the Civil Rights Movement emerged the legislative
impetus that mandated the same standards being taught to and learned by each student. In 1975
the Federal legislators enacted the Individuals with Disabilities Education Act (IDEA) mandating
not just equal access but also a minimum of equal outcomes in the expected learning;
subsequently, the legislators mandated a FAPE for each student in public education. The full
force of the mandate was renewed and modified IDEA (2004) and then finalized its inclusive No
Child Left Behind (NCLB) expectations (U.S. Department of Education, 2006). Both IDEA and
13
NCLB suggests a student's epistemology should drive a teacher's pedagogy; the priority of each
student's peculiarities impacting the individual's learning should inform the pedagogy.
These Federal expectations had to be measured by the states. Since the state of Georgia
had sought to comply with both IDEA and NCLB mandates by measuring eighth grade math
success through the state's CRCT math test, then educators must both identify and intervene for
those students thought to be at risk of not meeting the standards (Georgia State Board of
Education, 2001; Henry, Rickman, Fortner, & Henrik, 2005; Livingston & Livingston 2002;
Mordica, 2006). In fact, students must pass the math test during the gateway grades of third
fifth, and eighth to be advanced to the next grade. FAPE was measurable through the standards
based CRCT math assessment. Retention was the immediate consequence of not passing the
eighth grade math test.
Assessment determined promotion to high school. Although the assessment had
implications for FAPE effectiveness, the limited budgets impacted the efficiency, time on
task. Relevant learning theories are informative and directional for supporting students at risk of
failing the assessment. Although Piaget's stage of cognitive development suggests that learning
is possible, historically, there was and still is much controversy over differentiated learning. The
impetus of the debate has been identified in two opposing articles and the collection of responses
found in blogs. Brenneman (2015) compiles the blogs of the debaters weighing in on the two
published pieces, Jim Delisle's "Differentiation doesn’t work," and Carol Ann Tomlinson's
timely response, "Differentiation, does, in fact, work."
While educators debate over effective learning strategies (Munk, Gibb, & Caldarella,
2010), what is most important is that educators both identify and address the learning of students
at risk of failing the CRCT. Identification and implementation of student specific interventions
14
are not a suggestion, but rather, they are expedient in meeting the student's FAPE (Georgia
Department of Education, 2008). While effective learning strategies are historically and
currently debatable (Baker, Rieg, & Clendaniel, 2006; Flores & Kaylor, 2007), all would concur
that time on task and repetitive learning target experiences must be considered for success in
math interventions (Axtell, McCallum, Mee Bell, & Poncy, 2009).
Conceptual Framework/Theory Overview
U.S. education’s proclivity has reproduced a specific socioeconomic dominant group,
the middle class. By perpetuating the middle class, others are underappreciated and
undereducated. Sociologists, Bourdieu and Passeron, have “developed a theory of reproduction
in education” that identifies “achievement gaps” as actually socioeconomic “opportunity gaps”
(Huddleston, 2014, p. 5). Lareau (2003) has suggested that socioeconomic inequities contribute
to academic gaps for the non-dominant groups. U.S. education has reproduced a middle class
model consistent with certain dominate socioeconomic values, resources, and skills that exist in
the local middle class; teachers teach to the middle to perpetuate these educational outcomes.
The Bourdieu and Passeron (1990) theory suggests that non-dominant students need equitable
opportunities to address the achievement (opportunity) gaps.
These equitable opportunities are possible. The theoretical base for this study includes
Piaget’s stage of cognitive development, specifically, the concrete operational theory, the
constructivist theory, and the behaviorist theory. While Piaget's stages identifies the student's
cognitive maturity, suggesting that the student can learn, the constructivist and behaviorist
theories connect the learning with both the learner and the environment (Ertmer & Newby, 2013;
Sezer, 2010). What is most relevant is the opportunity and motivation to learn (Ertmer &
Newby, 2013).
15
Piaget’s concrete operational theory suggests that cognitive development has matured in
the learner to the point that the learner can both learn math rules and then apply those rules to
physically perceived objects (Atherton, 2013). While behaviorists are creating extrinsic support
through a positive and rewarding learning environment, constructivists suggest that the learners
can take that information and connect it in such a way that implementation is evident by what the
learner is able to construct and communicate (Ertmer & Newby, 2013). Thus, both direct
instruction and time on task are expedient for learning the math rules, moving the rules from
short term memory to long term memory (Barbash, 2012; Ertmer & Newby, 2013). Also, most
will need manipulatives and visuals in implementing the rules learned; once again, time is
indicative in learning, implementation, and retention. Piaget, behaviorists, and constructivists
suggest that learning is made through connections (Ertmer & Newby, 2013). When middle
schoolers make cognitive connections with the new learning, learning occurs with relevance to
the learner's environment (Schrank & Wendling, 2009).
When students do not connect with learning the math standards, then remedial
interventions must become student specific. In fact, specific learning disabilities have been
targeted with specific strategies that support each student's learning (Schrank & Wendling,
2009). Both the constructivist and behaviorist theories suggest that remediation is possible
(Ertmer & Newby, 2013; Schrank & Wendling, 2009; Sezer, 2010).
Problem Statement
The problem is that math proficiency, or the lack of it, empowers or retains a student's
progress in the American public education system (Bicknell, 2009; Gordon, 2007; Klein, 2003).
Educators were concerned with the Nation’s 2005 Report Card indicating that 31% of eighth
grade math students lacked proficiency (U.S. Department of Education, 2006; National Academy
16
of Sciences, National Academy of Engineering, and Institute of Medicine, 2005). Eighth graders
in the state of Georgia are retained if they are not proficient in math as measured by the state’s
CRCT (Henry et al., 2005; Livingston & Livingston, 2002; Mordica, 2006). Historically,
proponents of retention (Greene & Winters, 2007; 2009; Owen & Ranick, 1977) suggested that
more time on task through retention should be considered; however, with some, the negative
implications of retention eclipsed retention and empowered social promotion (Anagnostopoulos,
2006; Winters & Greene, 2012). McCombs, Kirby, and Mariano (2009) suggest that more time
on task and more effective learning can support students in becoming successful without
retention or social promotion. However, no research has examined the effectiveness of an
intensive nine-week remedial math connection (MC) class to improve math scores on the
Georgia CRCT eighth grade math test.
Given the low proficiency in math among eighth grade students, there is a need to
intentionally offer equitable education to bridge the academic gaps for eighth grade students at
risk of failing (Jitendra, 2013; Malmgren, McLaughlin, & Nolet, 2005; Munk et al., 2010;
Schrank & Wendling, 2009). The academic gap between expected math proficiency and current
levels of performance suggests a future inequity in limiting the access to the high school math
curriculum. Therefore, it is imperative that compensatory interventions are identified and
implemented. While most agree that there is a need for both more time on task and more
effective means for learning, the intentional, equitable amount of time has not been addressed for
bridging the math gap that would give eighth graders full access to the high school math
curriculum. While the amount of time required (e.g., summer school, retention for another year,
after school math, two math classes, more allotted time for the remediation) has been debated
(Jacob & Lefgren, 2004; 2009; Matsudaira, 2007; Roderick & Nagaoka, 2005), the literature has
17
not considered students receiving an additional MC class for 55 minutes a day for 45 days. No
research to date has addressed the impact of an additional MC class for 55 minutes a day for 45
days in order to bridge the inequity of math gaps for eighth grade students that are at risk of
failing state mandated math assessments. The problem is that lack of math proficiency precludes
a student's progress in the American public education system, specifically, in the state of
Georgia.
Purpose Statement
The purpose of this quantitative quasi-experimental study is to see if an intensive nine-
week remedial MC class (based upon the eighth grader’s seventh grade math CRCT scores) can
significantly increase scores. The dependent variables are the AIMSweb posttest scores in math.
The independent variable is where one group participates in the nine-week remedial math
connection class and the other group will not. Those who are in the remedial math connection
class scored an 820 or lower on the previous year’s CRCT math scores or the present year’s math
teacher’s recommendation that the student is at risk of failure. A covariate will be used, which
will be the AIMSweb pretest scores. This will be used to control for differences in AIMSweb
pretest scores between the control and treatment groups. Students deemed at risk will be
preassigned to a treatment group based upon the previous year’s (2010) CRCT math scores (820
or lower), or the student’s 2010-2011 math teacher’s perception of the student being at risk of
failing the CRCT.
Significance of the Study
The significance of this study is the consideration of the MC class being both an efficient
and effective means of “catching kids up” (Beatty, 2012a; Takanishi, 2012). Most research has
suggested that more time on task is needed for students who are behind (Beatty, 2012b; Span,
18
2012). Concomitantly, as states move toward the Common Core State Standards (CCSS)
curriculum (or similar curriculum), pedagogy must address content, process, and assessment in
light of each student’s epistemology. Also, the fiduciary responsibility, with both fiscal and time
constraints, suggests that schools should implement an efficient and cost-effective program that
provides a FAPE for each student.
This study seeks to better understand how to enable students to access the curriculum.
Access to the future curriculum requires skills in both math and reading at each grade level.
Remediation is possible (Clements & Sarama, 2011; Engel, Claessens, & Finch, 2013; Shapiro,
2011). Although students learn differently, all students can learn (Claessens, Engel, &
Curran, 2013). Behaviorists’ and constructivists’ perspectives best supports this quasi-
experimental design based upon the research of acceleration and remediation (Ertmer & Newby,
2013; Sezer, 2010; Thompson, Thompson, & Thompson, 2002). Consequentially, this theory
suggests that differentiation of learning must match the need of the student (Wiles et al., 2006).
The constructivist theory supports both Piaget’s cognitive aspect (Atherton, 2013) and
Vygotsky’s (1978) social aspect in relation to eighth grade learners.
This study considers the literature suggesting that more time spent on math over and
above what is allotted in the classroom can be effective (Hall, Strangman, & Meyer 2011; Stone,
Engel, Nagaoka, & Roderick, 2005). This additional time, nine weeks, both remediates and
accelerates, since the curriculum is based on an overview of Georgia’s standards based eighth
grade math curriculum as assessed through the CRCT. Research has suggested that time frames
such as summer school, retention, after school programs, and double math classes have impacted
some learners (Jacob & Lefgren, 2004; 2009; Matsudaira, 2007). Economically, some school
systems cannot afford these programs; however, an extra nine-week math class may be more
19
affordable if it can become efficient. An overview of the literature suggests a lack of research on
an intense nine-week MC class (Foegen, Jiban, & Deno 2007; Gersten et al., 2009).
Research Question
The research question for this study was:
RQ1: While using the AIMSweb pretest scores as a control variable for previous math
achievement, will at-risk eighth grade students who attend an intensive nine-week math
connection class have statistically significant different mean scores as measured by the
AIMSweb posttest when compared to students who do not receive the compensatory math
instruction?
Null Hypothesis
The null hypothesis for this study was:
H01: While using the AIMSweb pretest scores as a control variable, at-risk eighth grade
students who attend an intensive nine-week math connection class will not have statistically
significant different mean scores as measured by the AIMSweb posttest when compared to
students who do not receive the compensatory math instruction.
Definitions
1. AIMSweb - A nonmultiple choice test format that helps evaluate a student’s math
performance; some schools incorporate this as a part of an identification of a student’s
need for interventions (AIMSweb, 2009a).
2. Common Core State Standards (CCSS) - Curricular standards that most states have
adopted to implement (Porter, McMaken, Hwang, & Yang, 2011).
3. Criterion Referenced Competency Test (CRCT) - The state of Georgia’s annual
summative assessment to measure a student’s status within a group (did not meet, meets,
20
exceeds) based upon prescribed standards (Georgia Department of Education, 2008,
2009, 2010, 2011).
4. Free Appropriate Public Education (FAPE) - The primary aim of Civil Rights in
education, including No Child Left Behind, a Response to Intervention, and Georgia’s
new assessment program, is that each student has an equitable opportunity to an
education (U.S. Department of Education, 2007b).
5. Learning-Focused Schools (LFS) - A systemic pedagogy that seeks to teach a student the
way the student learns best; the pedagogy considers the student’s epistemology
(Thompson et al., 2002).
6. Math Connection (MC) - An extra nine-week small group, learning-focused math class
that both remediated and accelerated the eighth grade CRCT math standards for students
identified as at risk. Conversely, not-at-risk students were placed in other connection
classes that were not math related (Williams, 1996).
7. Math Difficulty (MD) - This includes students who have been officially identified with a
disability and other students who manifest math difficulties (Powell, Fuchs, & Fuchs,
2013).
8. No Child Left Behind (NCLB) - Legislation passed by Congress in 2001 to monitor
student achievement data (Linn, Baker, & Betebenner, 2002).
9. Response to Intervention (RTI) - A systemic approach to identifying a student’s level of
need and specific ways to support a student in meeting the need; it includes formative
assessment through both progress monitoring and new ways of teaching for the way a
student learns (Georgia Department of Education, 2008).
21
CHAPTER TWO: LITERATURE REVIEW
Introduction
This chapter introduces the literature review, connects the relevant theoretical framework,
provides an overview of the literature, and summarizes the literature. In searching for relevant
literature, certain keywords were used. They included: summer school, retention, time on task,
after school, students at risk, remediation, acceleration, student achievement, middle school
math, NCLB, RTI, CRCT, and high stakes testing. From these queries, the most recent and
relevant research bibliographies were also considered.
High stake’s testing in the state of Georgia utilizing the annual CRCT has had students
respond in multiple choice format. However, the proposed new testing suggests that the
assessment will include constructed-response format. Conversely, as early as Linn, Baker, and
Betebenner (1991), disparity and test inequalities between the two testing formats of multiple
choice answers (MCA) and the open end question, the precursor to the constructed response
(CR), have been argued. However, that disparity has greater implications for low-achieving
students, including those with disabilities and those who may be in the process of being
identified with a disability, and students at risk of failing. In fact, Powell’s (2012) research
argued for MCA as a testing accommodation for students with a learning disability in math.
Powell suggested that the third grade math students “responding in the multiple-choice format
had a significant advantage over students answering in the constructed response format” (p. 3).
As NCLB (U.S. Department of Education, 2002) has left the shadow of high stakes
testing on education in Georgia, the purpose of this research was to consider the academic
intervention of remediation and acceleration to improve the eighth grade outcomes of both the
CRCT and a corollary test, the AIMSweb. These scores could be quite suggestive for the state’s
22
math CRCT scores at a middle school in Georgia as impacted through the treatment of an extra
math connection (MC) class. The administrators of NCLB consider the student’s performance
on the standardized math test as an acceptable assessment of a student’s proficiency of the state’s
standards (Marsh, Pane, & Hamilton, 2006). Caldwell (2008) argued that these standardized
tests measure three things: “what a student comprehended and learned,” “the student’s
socioeconomic status (SES),” and “the student’s inherited academic aptitude” (p. 183). Because
states will continue to measure the success for each student, it is imperative to intervene for low
achievers in both an effective and a cost and time efficient manner (Petscher et al., 2012).
Developing cognitive strategies to help at-risk middle school students with their math has
become a growing concern (Krawec, Huang, Montague, Benikia, & Melia de Alba, 2012). Since
the 1970s, time on task and mastery of content has been discussed by educators. John Carroll set
the stage with his 1963 paper, A Model of School Learning (Carroll, 1989). Although the debate
continues, some do not value the necessity of time exclusivity. “The study findings indicate that
content coverage positively and significantly influences pupil achievement if it is the only
predictor. There is no support for the hypothesis that time-on-task predicts achievement”
(Oketch, Mutisya, Sagwe, Musyoka, & Ngware, 2012, p. 31). However, the quality of time has
been purported to be significant (Marzano, Pickering, & Pollock 2001; Stonehill et al., 2011). In
fact, Marzano et al. (2001) has suggested four qualifiers for most effective curriculums: explain,
model, guide a practice, and allow for independent practice. This is consistent with Piaget's
cognitive development theory, especially with the learner constructing meaning through both
assimilation and accommodation (Atherton, 2013). Even with specific interventions, the process
should be both effective and efficient (Redd et al., 2012). “It is clear that a limited number of
23
studies evaluate the effectiveness of an intervention with regard to the amount of instructional
time needed to implement the intervention” (Bramlett, Cates, Savina, & Lauinger, 2010, p. 114).
As schools have attempted to address interventions for students at risk and better ways
to include students with special needs in the general classroom, efficiency and effectiveness have
become paramount (Cosier, Causton-Theoharis, & Theoharis, 2013). There is a need for
teachers to consider individualistic epistemologies and rethink the teacher’s pedagogy. This can
produce diverse interventions while seeking effective strategies that consider the individual
student’s epistemology (Gersten et al., 2009; Nomi & Allensworth, 2013). In their study, Methe,
Kilgus, Neiman, and Riley-Tillman (2012) looked at the math functions of addition and
subtraction and analyzed the effect size through a meta-analyses of 47 effects in 11 studies
(effect sizes ranged from 0.59 to 0.90). “Variables that appeared to moderate the effects were
student age, time spent in intervention, and intervention type” (Methe et al., 2012, p. 230). They
identified a need for future research “in basic arithmetic and rigorous experiments” for the
purpose of establishing “an evidence base that accurately characterizes intervention
effectiveness” (Methe et al., 2012, p. 230).
Historically, for interventions to be implemented, at least two things have become
apparent: teachers need to be able to quickly identify the at-risk students and then discover
effective strategies. Krawec (2013) had considered the epistemologies of three groups of
students: learning disabled students (LD, n = 25); low-achieving students (LA, n = 30); and
average-achieving students (AA, n = 29). What was most distinguishing was not that there were
inequities in the groups’ ability to restate what the math problem was asking, but rather “the
effect of visual representation of relevant information on problem-solving accuracy was
dependent on ability; specifically, for students with LD, generating accurate visual
24
representations was more strongly related to problem-solving accuracy than for AA students”
(Krawec, 2013, p. 80). Krawec’s research suggests that teachers need to understand the student’s
proclivities for learning. A better informed epistemology is quite suggestive for a differentiated
pedagogy, especially for forming approaches that what will allow the student to learn in different
ways and on different days of complex content (Gamble, Kim, & An, 2012). Also, Yell and
Walker (2010) suggested that, legally, these at-risk groups must be educated through effective
interventions.
Theoretical Framework
Sociologists, Bourdieu and Passeron, have “developed a theory of reproduction in
education” that identifies “achievement gaps” as actually socioeconomic “opportunity gaps”
(Huddleston, 2014, p. 5). Lareau (2003) has suggested that socioeconomic inequities contribute
to academic gaps for the non-dominant groups. U.S. education has reproduced a middle class
model consistent with certain dominate socioeconomic values, resources, and skills that exist in
the local middle class (Chapman, Tatiana, Hartlep, Vang, & Lipsey, 2014); teachers teach to the
middle to perpetuate these educational outcomes. The Bourdieu and Passeron (1990) theory
suggests that non-dominant students need equitable opportunities to address the achievement
(opportunity) gaps.
These equitable opportunities are possible. The theoretical base for this study includes
Piaget’s stage of cognitive development, specifically, the concrete operational theory, the
constructivist theory, and the behaviorist theory. While Piaget's stages identifies the student's
cognitive maturity, suggesting that the student can learn, the constructivist and behaviorist
theories connect the learning with both the learner and the environment (Ertmer & Newby, 2013;
25
Sezer, 2010). Practically, what is most relevant, is the opportunity and motivation to learn
(Balfanz & Byrnes, 2006; Ertmer & Newby, 2013).
Piaget’s concrete operational theory suggests that cognitive development has matured in
the learner to the point that the learner can both learn math rules and then apply those rules to
physically perceived objects (Atherton, 2013). While behaviorists are creating extrinsic support
through a positive and rewarding learning environment, constructivists suggest that the learners
can take that information and connect it in such a way that implementation is evident by what the
learner is able to construct and communicate (Ertmer & Newby, 2013). Thus, both direct
instruction and time on task are expedient for learning the math rules and moving those rules
from short term memory to long term memory (Barbash, 2012; Ertmer & Newby, 2013). Also,
most will need manipulatives and visuals in implementing the rules learned; once again, time is
indicative in learning, implementation, and retention. Piaget, behaviorists, and constructivists
suggest that learning is made through connections (Ertmer & Newby, 2013). When middle
schoolers make cognitive connections with the new learning, learning occurs with relevance to
the learner's environment (Schrank & Wendling, 2009).
When students do not connect with learning the math standards (Dawn & Mendick,
2013), then remedial interventions must become student specific for each student. In fact,
specific learning disabilities have been targeted with specific strategies that support each
student's learning (Schrank & Wendling, 2009). Both the constructivist and behaviorist theories
suggest that remediation is possible (Ertmer & Newby, 2013; Schrank & Wendling, 2009; Sezer,
2010).
Remediation is possible (Shapiro, 2011). Although students learn differently, all students
can learn. While the behaviorist seeks to construct an environment for motivating the learner,
26
the constructivist seeks to encourage the learner to construct meaning from the learning
environment (Wiles, Bondi, & Wiles, 2006). Constructivist research suggests that math
achievement gaps among eighth grade students can be addressed through acceleration and
remediation (Thompson et al., 2002). Consequentially, this theory suggests that differentiation
of learning must match the need of the student (Wiles et al., 2006). The constructivist theory
purports both Piaget’s cognitive aspect and Vygotsky’s (1978) social aspect (Atherton, 2013).
Constructivists use the social interaction of the teacher with the student to create a
learning environment that supports the student’s learning by implementing a social cognitive
process that is both suggestive and directional—modifying and adapting the learning to meet the
student’s most specific needs—which subsequently empowers the learner by creating and
perpetuating a sense of self-efficacy (Bandura, 1997; Marzano, 2003; Posner & Rudnitsky,
2006).
Achievement gaps suggest that compensatory strategies must be implemented to close the
gaps (Sobel & Taylor, 2006); if not, Judge and Watson (2011) suggested that the gap increases
with each grade. However, with extra time on task, the diminishing of the gap is not guaranteed
(Bennett et al., 2004). In fact, the meta-analysis of Lauer et al. (2006) suggested that for the time
on task to effectively diminish the gap, the process must be student-specific based upon the
content complying with the state standards.
Theoretically, each child can learn (Linn et al., 2002), learning is measurable, and if a
student fails to learn, then academic intervention is essential. Therefore, the pedagogy and
epistemology must be realigned. Research has suggested that these assumptions are possible
through academic intervention (Hattie, 2009, 2011). Also, specific academic intervention is
plausible if this realignment considers the theory of multiple intelligences (Gardner, 1999) and
27
differentiation (Tomlinson, 1995a) that is designed to meet the student’s needs (Hall et al.,
2011). Practically, the luxury of time and money has been deprecated in the shadow of budget
constraints. Therefore, there is an urgent need to be both efficient and effective to provide
compensatory education for students who are at risk.
Related Literature
Compensatory Education
Compensatory education is possible (Krawec, 2013; Stone 1998; Valencia, 2012;
Walston & McCarroll, 2010). Efficient and effective approaches must be identified and
implemented (Shapiro, 2011). Hattie’s (2009, 2011) meta-analysis of previous studies suggested
that the most effective influences can be identified. These previous studies researched influences
impacting the learning of students. Hattie’s meta-analysis considered millions of students.
Hattie systemically considered different influences on learning for over a period of 15 years; he
was able to consider the data and establish a mean score (.40). The effect score assigned to each
influence was based upon his meta-analysis of the data. Hattie systemically categorized these
influences as student, home, school, curricula, teacher, and teaching and learning approaches;
he then ranked the specific impact of each influence based upon the effect size on student
learning relative to the mean.
When a student does not learn, the epistemology of the student needs to be considered
(Foegen et al., 2007; Petscher et al., 2012; VanDerHayden & Burns, 2009). Once the learner’s
proclivities are better understood, the learner’s specific pedagogy can better address academic
gaps by seeking to individualize the learning during remediation. What a student knows,
including skill sets, are quite suggestive for pedagogy and interventions (Clements & Sarama,
2011; Engel et al., 2013).
28
Pedagogy and Epistemology
Pedagogy and epistemology are relevant (Harlacher, Nelson, & Sanford, 2010). When a
student does not learn from the pedagogy, it is expedient to discover how to facilitate learning in
a manner in which the student learns best (Tomlinson, 1995b). Using the student’s epistemology
to better address the ways and methods to support the learner is imperative; the teacher can better
customize the differentiation of the curriculum (Dougherty et al., 2015). Perhaps specific
learning style, small group, and direct instruction are just a few of the approaches that might
improve the learning process (Rickles, 2013; Schatschneider et al., 2008). Learning that
considers both the student’s interest and the scaffolding of the learning based on how a student
learns best suggests improvement. Concomitantly, when both teacher and student succeed, then
both teacher efficacy and student efficacy emerge into a collective student-teacher efficacy
(Brown, 2010; Brown, Benkovitz, Muttillo, & Urban, 2011; Munk et al., 2010).
The ongoing challenge has been and remains to be how to best maximize the time needed
to facilitate the bridging of the academic gaps (Burns & Gibbons, 2012; Goddard, Hoy, &
Woolfolk-Hoy, 2004). Thus, if given enough time to intervene, and if the intervention is
individualized, then perhaps the learning is impacted through teacher efficacy (Guskey &
Passaro, 1998; Hall et al., 2011; Ross, 1994). Subsequently, this additional time for intervention
could substantially impact a student’s summative learning outcome when aligned with the
curriculum-based standards as measured by the CRCT (Silva, 2007; Siwatu, Polydore, & Starker,
2009; Siwatu & Starker, 2014; Tucker et al., 2005; Vaughan, 2002). With standards-based
content, a differentiated process, and a relevant summative assessment, there is no need for a bell
curve distribution of grades that expects a certain amount of failures; however, math proficiency
suggests a pedagogy with a proclivity toward the student’s epistemology.
29
Proficiency at each academic grade level suggests that the United States can be globally
competitive (Porter et al., 2011). The need to globally compete impacts the American public
education system (Bicknell, 2009; Gordon, 2007; Klein, 2003); one 2005 report indicated that
about two thirds of eighth grade math students lacked proficiency (National Academy of
Sciences, National Academy of Engineering, and Institute of Medicine, 2005). Subsequently,
the appropriation of the COMPETE Act, an allocation by the federal government of $33.6 billion
to address math and its interdisciplinary deficits (Bicknell, 2009; Committee on Science and
Technology, 2007), suggests that each institution must consider both the cost and the correlation
of effectiveness in addressing the math gap.
Because gaps do exist, resulting in low achievers, it is therefore the responsibility of
educators to provide a FAPE (Valero, 2012). To reverse the trend of a low achiever, educators
must respond (Krawec et al., 2012). The variables of causation must be considered to reverse the
trend (Blankstein, 2013; Dembosky, Pane, & Christina, 2006). Fischer and Frey (2007) found,
“When comparing achievement data at aggregate levels, differences based on ethnicity and race,
language, and gender are obvious” (p. 10); however, Goddard et al. (2004) argued for a greater
significance in the faculty’s collective perception (Mojavezi & Tamiz, 2012). The collective
perception of being able to effect change is much more consequential for the student’s outcomes
than the student’s socioeconomic status (Goddard et al., 2004; Hattie, 2011). Although multiple
causation including efficacy, socioeconomic status, ethnicity, religious beliefs, and/or poor
learning environment exists, most would agree that a quintessential underlying factor in the
diminishing of the gap is time (Bennett et al., 2004; Redd et al., 2012); however, effecting
change requires an effective use of the time. It seems that both time and cost force educators to
rethink the learning environment and how to use formative assessment to inform summative
30
assessment (Fuchs et al., 2008). Although other schools have addressed the problem of low
achievers (Duffrin & Scott, 2008), what distinguishes this research is the limited treatment that
the student will receive in light of time and cost restraints to reverse the low achievement as
demonstrated in the pre and posttest outcomes of both the CRCT math test and a corollary test,
the AIMSweb.
While many educators are highly focused on state testing, it is important to consider that
over the course of a year, teachers can incorporate many opportunities to assess how students are
learning and then use this information to make beneficial changes in both time on task and the
learning process (Foegen et al., 2007; Petscher et al., 2012; VanDerHayden & Burns, 2009).
This diagnostic use of assessment to provide feedback to teachers and students over the course of
instruction is called formative assessment. It stands in contrast to summative assessment, which
generally takes place after a period of instruction and requires making a judgment about the
learning that has occurred, for example, through grading or scoring a test or paper (Boston, 2002,
n.p.).
Formative Assessment
Formative assessment, which informs summative assessment, could benefit the low
achiever (Phelan, Choi, Vendlinski, Baker, & Herman, 2011). However, teachers need help in
assessing and monitoring a student. Teacher perception of a student’s academic achievement is
most important. Research has suggested that teachers have relative accuracy. Although the
research sample of Eckert, Dunn, Codding, Begeny, and Kleinmann (2006) was quite limited, the
results were quite suggestive for math remediation. While the teachers were limited in assessing
a student’s ability to perform math functions, their strength was assessing addition. Conversely,
31
they were less likely to be able to assess a student’s math level as either mastery, instructional, or
functional.
The social-cognitive theorist implemented formative assessment in the scaffolding of
each learner (Kagan, 1994; Marzano, 2003; Vygotsky, 1978). Formative assessment suggests an
early identification of an academic gap as opposed to the summative assessment. The
summative assessment identifies a much larger gap that is most difficult to bridge in light of the
need to learn new material. The larger the gap, the greater the challenge (Nomi & Allensworth,
2011).
As past approaches of remediation often resulted in students falling even more behind,
Learning Focus Schools proposed a program of remediation and acceleration for a low achiever
(Thompson et al., 2002). The goal was to “catch up” the low achiever through a continuity of
remediation and acceleration; remediation aimed to bridge the gaps and acceleration sought to
build confidence—namely, what one should have learned and a preview of what one should
learn next, respectively. This form of acceleration is not preteaching; rather, it functions like a
trailer for an upcoming movie; it activates prior knowledge, and introduces key concepts and
vocabulary consistent with Piaget’s cognitive development theory.
Remediation and acceleration connects the learning and makes it visible (Hattie, 2011).
The key is the formative assessment that scaffolds the learner through both remediation and
acceleration. Succinctly stated:
When teachers know how students are progressing and where they are having trouble,
they can use this information to make necessary instructional adjustments, such as
reteaching, trying alternative instructional approaches, or offering more opportunities for
32
practice. These activities can lead to improved student success. (Boston, 2002, n.p.;
Fullan, Hill, & Crevola, 2006)
Even though in some learning environments “an enormous proportion of daily assignments are
simply never assessed—formally or informally—and no evidence exists by which a teacher
could gauge or report on how well students are learning essential standards” (Schmoker, 2006, p.
16), a school can intentionally implement formative assessment (Fisher & Frey, 2007), thus
creating a climate and culture change that significantly impacts the low achiever (Lukas &
Beresford, 2010).
Although the restructuring process to implement a formative assessment that can inform
summative assessment may result in different frameworks, Fisher and Frey (2007) identified four
essentials for the framework (p. 12). These four essentials are:
Aligning with enduring understandings (Wiggins & McTighe, 2005).
Allowing for differentiation (Tomlinson, 1999).
Focusing on gap analysis (Bennett et al., 2004).
Leading to precise teaching (Fullan et al., 2006).
The framework is quite suggestive. The premise of helping the low achiever
substantiates the fundamental assumption of NCLB that each student can learn; concomitantly,
learning is not innate. In fact, this framework of learning seems to be most effective; “effect
sizes ranged between .4 and .7, with formative assessment apparently helping low-achieving
students, including students with learning disabilities, even more than it helped other students
(Black & William, 1998)” (Boston, 2002, n.p.). In fact, low achievers often lose focus and
motivation, thinking that learning is innate. Conversely, Boston argued that the formative
feedback informs the learners of “any gaps that exist between their desired goal and their current
33
knowledge, understanding, or skill and guides them through actions necessary to obtain the goal”
(Boston, 2002, n.p.).
Formative assessment both identifies the error and informs the pedagogy in a non-
threatening manner. It encourages students to take risk without academic failure. This seems to
enhance the learning for the low achiever as he/she is rewarded for “effort rather than be doomed
to low achievement due to some presumed lack of innate ability” (Boston, 2002, n.p.). This new
approach that includes formative assessment, consistent with the intent of NCLB, requires a
paradigm shift; the learning environment must be reconstructed for the specific needs of each
learner in anticipation of creating the construct of self-efficacy in each learner. The belief is that
formative assessment helps support the expectation that all children can learn to high
levels and counteracts the cycle in which students attribute poor performance to lack of
ability and therefore become discouraged and unwilling to invest in further learning
(Boston, 2002, n.p.).
When formative assessment is being aligned with the state standards, schools need to
address sociopolitical inequity in the organization of mathematics (Valero, 2012). Inequities
must consider three objectives—“identify desired results,” “determine acceptable evidence,” and
“plan learning experiences and instruction” (Wiggins & McTighe, 2005, p. 9)—that essentially
integrated the state CRCT standards (Fullan et al., 2006; Noell, 2005; Posner & Rudnitsky,
2006).
Grouping is relevant to the uniqueness of the student. The strategies of both individual
and small group learning (Brown et al., 2011; Smith & Bell, 2014) are quite suggestive in
addressing inequities. Tombar and Borich (1999) “urged educators to make classroom learning
more of a joint cognitive venture among all classroom participants than a solitary enterprise” (p.
34
189). Formative assessment must consider the uniqueness of the student. In fact, “the ways
people communicate and construct meaning…depend on social interaction and cultural context”
(Maker & Schiever, 2005, p. 293), and the formative assessment “results indicate that many
teachers find peer and self-assessment useful and that there is potential for greater classroom
applicability” (Noonan & Duncan, 2005, n.p.). The consensus seems to be that, “There is little
question that such grouping arrangements lead to higher degrees of complex learning in
comparison to whole-group teaching methods” (Tombar & Borich, 1999, p. 189).
In addition, small group affords a systemic formative assessment (Cusumano, 2007;
Fredriksson, Ockert, & Oosterbeek, 2013) that can better inform both the construct of learning
and the cognitive outcome (Petscher et al., 2012). Subsequently, systemic implementation of a
curriculum-based measurement (CBM) potentially diminishes the gap through early
interventions (Hosp, Hosp, & Howell., 2007; Jiban & Deno, 2007). Although Espin, Scierka,
Skare, and Halverson (1999) “examined the criterion-related validity of curriculum-based
measures in written expression” (p. 5) of 147 tenth graders, it was the formative assessment
approach of the CBM that was most useful. Espin et al. (1999) suggested this approach as “a
systemic procedure for monitoring students’ progress in an academic area and making
instructional decisions” (p. 5). In addition, “Underlying this approach is the value of economic
efficiency—that is, promoting greater output with no increase in expenditure” (Ladd & Walsh,
2002); concomitantly, some students’ cognitive performances are impacted when passing the test
is determinative for moving up to the next grade level, as seen in high stakes testing (Roderick,
Bryk, & Jacob, 2002).
Formative assessment with CBM (Shinn, 2008; VanDerHayden & Burns, 2009) is both
informative and directional for instruction (Safer & Fleischman, 2005). Formative assessment
35
“can lead to increased precision in how instructional time is used in class and can assist teachers
in identifying specific instructional needs” (Espin et al., 1999, p. 48). How does this inform
pedagogy? Pedagogically, the final report of the National Mathematics Advisory Panel (2008)
suggested formative assessment as a tool to both inform and then give direction for future
learning by stating, “Teachers’ regular use of formative assessments improves their students’
learning, especially if teachers have additional guidance on using the assessment results to design
and individualize instruction” (p. 47). When a student has lost some confidence, formative
assessment can reestablish success and a blending of confidence with self-efficacy. In fact, “For
struggling students, frequent (e.g., weekly or biweekly) use of these assessments appears
optimal, so that instruction can be adapted based on student progress” (p. 47). This scaffolding
of learning suggests a need to research the “specific tools and strategies”; however, the pedagogy
includes tutoring, computer assistance, and “a professional (teacher, mathematics specialist,
trained paraprofessional)” (National Mathematics Advisory Panel, 2008, p. 47).
Yeh (2010a; 2010b) considered 22 approaches that have been implemented to improve
student achievement. He suggested that the rapid assessment of student daily and weekly
achievement allows for the individualizing of the learning. His findings showed that rapid
assessment with immediate adjustments to accommodate the epistemology of the learner is most
cost-effective, more so
than comprehensive school reform (CSR), cross-age tutoring, computer-assisted
instruction, a longer school day, increases in teacher education, teacher
experience or teacher salaries, summer school, more rigorous math classes, value-
added teacher assessment, class size reduction, a 10% increase in per pupil
expenditure, full-day kindergarten, Head Start (preschool), high-standards exit
36
exams, National Board for Professional Teaching Standards (NBPTS)
certification, higher teacher licensure test scores, high-quality preschool, an
additional school year, voucher programs, or charter schools. (p. 38)
Epistemologically, with the National Mathematics Advisory Panel (2008) suggesting
both an efficient and effective means of helping “students with learning disabilities (LD) as well
as low-achieving (LA) students” (p. 48), two presuppositions to learning are differentiation and
immediate formative assessment. Systemically, the learner needs to be scaffolded based upon
that assessment (Geary, Hoard, Nugent, & Byrd-Craven, 2009; Nomi & Allensworth, 2013).
Concomitantly, the underlying strategy for both LD and LA is to include opportunities for
students to engage in math talk, asking questions and working through the process of solving the
problem out loud (Rosenzweig, Krawec, & Montaque, 2011). Although quite engaging and
productive, math talk is merely a part of the essential kind of learning that should comprise
mathematics instruction (Methe et al., 2012). However, foundational math skills and math
concepts are quintessential for the math talk to occur with significance.
Internationally, the United States has fallen behind several countries academically.
While the achievement gaps are a global concern, the most pronounced proclivity within the
United States exists toward those of color, especially within low socioeconomic communities
(Ross et al., 2001; Valero, 2012). The research of Balfanz and Byrnes (2006) considered three
schools and four cohorts between fifth and eighth grades, where studies have found gains in
mathematics achievement. Balfanz and Byrnes identified that these schools were “implementing
whole-school reform models that incorporated research-based, proven curricula, subject-specific
teacher training and professional development, multiple layers of teacher and classroom support,
and school climate reforms” (p. 143). The researchers analyzed the data by applying a Binary
37
Logistic Regression model aimed at showcasing the factors that seemed to be a contributing
factor in closing the gap. This middle school research is quite suggestive:
We conclude that various student, classroom, and school-level factors are all key in
helping students to close the gap. WSR models, while often time and cost intensive,
address issues at all of these levels and may be more able to affect the achievement gap
than other, more simply implemented reforms. (p. 143)
The focus included three significant characteristics—attendance, behavior, and teacher
efficacy—which, when combined, resulted in 77% of the middle school students reaching grade
level. Both teacher and time on task can impact learning. Just as students need differentiation of
learning, which impacts pedagogy informed by each student’s epistemology (Siegler et al.,
2012), students also need different allotted time for learning different aspects of the math
curriculum (Atherton, 2013; Tomlinson, 1999). Since the mid-1990s, Tomlinson has advocated
differentiation, including mixed-ability classrooms (1995b) and being responsive to
epistemologies that are quite suggestive for differentiation of both pedagogy and time on task.
One time frame for all does not warrant expectations for mastering learning (Petscher et al.,
2012).
Differentiation and Disparity
Time on task has relevance to both pedagogy and each student’s epistemology (Durwood,
Krone, & Mazzeo, 2010; Krawec et al., 2012). However, what the teacher knows about the
subject content can also impact the learning of each student. In fact, the National Center on
Accessing the General Curriculum has proposed a chart that supports the above concerns for
differentiation (Hall et al., 2011). The chart consists of four categories that define
differentiation: mindset, ways to differentiate, epistemology, and instructional approaches. The
38
mindset for differentiation suggests that the learning environment can be most supportive with a
substantial curriculum that teaches up, uses groups and tasks that are both sensitive and flexible,
and utilizes assessments which are both informative and directional for both teaching and
learning (Claessens et al., 2013; Clements & Sarama, 2011).
The differentiated learning environment is characterized by different cognitive and
affective processes and products (Rosenzweig et al., 2011; Schellings & Broekkamp, 2011;
Shapiro, Keller, Lutz, Santoro, & Hintz, 2006; Wormeli, 2006). Each student’s epistemology
should be considered: the way the child learns best, the level of interest, and most of all, the
student’s ability to learn the specific task. The chart lists several instructional approaches:
“RAFTS; graphic organizers; scaffold reading; cubing; think-tac-toe; learning contracts; tiering;
learning/interest centers; independent studies; intelligence preferences; orbitals; complex
instruction; technology; web quests & web inquiry” (Hall et al., 2011, p. 5).
The differentiated classroom must be based upon some core beliefs, some core principles,
and some core practices. For learning to be differentiated, the teacher must believe the
following: all students can learn, diversity is normal, and failure is never an option (rather, it is
informative as the teacher constructs learning for each student to succeed). For learning to be
differentiated, the teacher must construct a positive learning environment, secure a core
curriculum, utilize assessments that both inform and give direction for both teaching and
learning, instruct based upon those assessments, and respond with flexibility. In addition, the
teacher must practice proactive prescriptive learning based upon each student’s epistemology,
instruction must be scaffold to meet a student’s need, teaching should challenge the student (the
zone of proximal development), assignments should be both affective and cognitively sensitive
39
and relevant, and “flexible grouping strategies (e.g., stations, interest groups, orbital studies)”
(Hall et al., 2011, p. 6) should be sensitive and relevant.
Teachers may make a significant difference. The value added by a teacher to a student
has been measured (Crane, 2002; Lissitz, 2014; Rivkin, 2007). Researchers have measured the
unit of growth attributed to a teacher’s impact for one year (Ross et al., 2001). This growth has
been based on standardized tests. Because of the focus of NCLB, researchers have often focused
on three subjects: reading, language arts, and mathematics (Stecher & Naftel, 2006; Sterbin,
2001; Stewart, 2006; Stronge & Tucker, 2000; Thum, 2002; 2003). In fact, some have argued
that instead of proficiency, NCLB should move toward unit of growth in determining AYP
(Viadero, 2006; Webster, 1998; Webster & Mendro, 1995). If learning is measurable, then
researchers will propose to measure and even reward teachers who add value to student learning.
But does the teacher need more resources to create the necessary value that is needed for at-risk
students (Krawec, 2013)? Thompson et al. (2002) posed the following questions:
What can a school do if a student is one to three grades, or more, behind in reading or
math? If teachers have students who are below grade level in their classrooms, what
tools or strategies are available that would actually accelerate a student's learning in order
to “catch him/her up?” (Thompson et al., 2002)
David Pupel’s (2001) book, Moral Outrage in Education, identifies disparity within
American society. He believes, “The energy that is created from the interaction of triumphalism,
timidity, and despair is surely entropic and hence only magnify our crises of poverty, inequality,
and polarization” (Pupel, 2001, p. 68). These inequities have caused Pupel to purport a vision of
reversal. Conversely, Pupel argued that we “reinstall our visions, dreams, and hopes for creating
a loving and just world and to recover our confidence in the human capacity to overcome the
40
obstacles to them” (p. 69). Equitable resources for a FAPE may be an essential key in reversing
the trend of teaching to the middle and enhancing the learning of each student.
The law requires that each student must be educated. The moral question must be asked,
“Who will control the content, process, and assessment of that education?” In America, the
resultant impact of education as stated by NCLB rests on the shoulders of the schools’ teachers
and leaders (Gonzalez, Frankson, & Shealey, 2008; Marzano, 2003; Tschannen-Moran & Barr,
2004; Waters, Marzano, & McNulty, 2003). As Annual Yearly Progress (AYP) ultimately
impacted the state and federal funding, myopically, each teacher and leader’s success became
correlated to each school making AYP (Finnigan & Gross, 2007). Although local schools have
much control over the implementation of the state standards, it is the state that ultimately
controls the content, process, and assessment; the school’s accountability under the NCLB and
its mandated AYP as the measure of success impact both local governance and funding.
Subsequently, this fiduciary responsibility inversely impacts the school when student groups,
those of lower socioeconomic status and/or those with disabilities, fail to meet AYP. To
continue with local control, not only must the dominant group of students meet a certain
minimum standard, but certain smaller student sub-groups must meet the minimum standard as
well.
Conversely, in the shadow of financial limitations and NCLB’s mandates, schools have
attempted to educate their various student groups in both an efficient and morally effective
manner. To help determine AYP in the state of Georgia, an annual assessment, the Criterion
Referenced Competency Test (CRCT), was administered to the middle school students. Also,
each eighth grader had to meet expectations on the CRCT in both reading and math to qualify for
41
promotion to the ninth grade. This two-edged sword placed pressure on each of the stakeholders
involved.
Although each state has a fiduciary responsibility to provide a FAPE, the content,
process, and assessment has changed much since the desegregation of Brown v. Board of
Education (1954; Donald, 2009; Ready, Edley, & Snow 2002). The desegregation practice has
become an underlying factor of the American Education system—no one is to be excluded (U.S.
Department of Education, 2007a). Although the history of compulsory, free public education has
been well documented, the empowering and limiting of a local school board’s actions have been
significantly impacted by both state and federal funding contingencies: Title I, socioeconomic
student equal access and opportunities; Title IX, female equal access and opportunities; and the
Individuals with Disabilities Education Act (IDEA), equitable access and opportunities for
students with special needs (U.S. Department of Education, 2007). Often, children with special
needs receive an Individual Educational Plan (IEP) that specifies an outline for their free access
to the general education curriculum. However, even with these desegregation supports, disparity
still exists (Domina, 2014; Donald, 2009; U.S. Department of Education, 2007a; Walston &
McCarroll, 2010).
Since the desegregation order of the landmark case of Brown v. Board of Education
(1954), inclusion of each child has become normative; however, academic disparity still exists
(Barton, 2004; Dougherty et al., 2015; Fram, Miller-Cribbs, Horn, & Lee 2007; McDowell,
Lonigan, & Goldstein, 2007; Rothstein, 2004). The desegregation order seems to have ignored
the low-income schools (Talbert-Johnson, 2004). Black and Hispanic low-income students
attend schools in which over two-thirds of students are identified as low income; conversely, less
42
than one-third are identified as low income within the context of White students (Martinez, 2012;
Silverman, 2004).
Disparity is identifiable between the Black and White groupings of students (Goodman,
2012). From 1971 to 2005, the trend has been to close the gap—the reading gap for 9-year-olds
improved from 44% to 29%, and for 13-year-olds, from 39% to 28%. In addition, after 1973, the
academic achievement gaps in math narrowed as follows: the 9-year-olds improved from 35% to
26%, and the 13-year-olds improved from 46% to 34% (Donald, 2009; Perie, Grigg, & Donahue,
2005). According to The National Center for Education Statistics (2007), the learning gaps from
1990 to 2005 diminished for Blacks and remained about the same for Hispanics. The fourth
graders’ gap of Black and White groupings diminished between the years 1990 and 2005,
decreasing from 32 to 26 points. Conversely, during that same time frame, the disparity
continued within the White-Hispanic groupings and remained at 20 points.
Concomitantly, the eighth graders reflected a similar diminishing between the Whites and
Blacks of 34 points and a disparity between Whites and Hispanics of 27 points. The gaps still
exist, and the National Education Association (2006) projects that the United States student
demographics will increase from one-third of minority students to one-half in 2025;
subsequently, coupled with the threat of school attrition, the paradigm must shift (Gonzalez et
al., 2008).
Each student who is required to attend school has the entitlement of a FAPE; however, in
the absence of a clear definition of appropriate, the courts have been interpreting FAPE
consistent with one’s civil rights—one must not “be excluded from the participation in, be
denied the benefits of, or be subjected to discrimination under any program or activity receiving
Federal financial assistance”; the FAPE must be “designed to meet their individual needs” (U.S.
43
Department of Education, 2007b, n.p.). The courts seem to advocate a view of moderation as
they have set some boundaries of extremes. The U.S. Supreme Court rejected maximizing “the
potential of each child with a disability” (Board of Education v. Rowley) and did not support
offering the mere minimum, de minimus (Walczak v. Florida Union Free School District);
instead, an appropriate education under the IDEA requires that the goal should seek student
progress while avoiding “significant regression” (La Morte, 2005, p. 333).
Proficiency, not excellence, is the goal of a FAPE (Cortes, Goodman, & Nomi, 2013;
2015; Domina, 2014). Math proficiency has impelled the American public education system in
the wake of failure (Bicknell, 2009; Gordon, 2007; Klein, 2003). The National Academy of
Sciences, National Academy of Engineering, and Institute of Medicine (2005) report, which
indicated that about two-thirds of eighth grade math students lacked proficiency, is more than a
pedantic concern. Subsequently, the interdisciplinary concerns precipitated the COMPETE
Act—an allocation by the federal government of $33.6 billion to address math and its
interdisciplinary deficits (Bicknell, 2009; Committee on Science and Technology, 2007).
Measurable Outcomes
Learning is measurable. As NCLB mandated that a school’s subgroups must meet the
preset minimal achievement standards as a criterion for meeting AYP, it is imperative for each
stakeholder to seek and support both an efficient and effective means to bridge the academic
gaps of each student. Potentially, the needs and desires of the majority can eclipse the needs and
desires of an individual or subgroup. This study, in the shadow of NCLB’s mandated AYP,
along with the state of Georgia’s self-imposed eighth grade mandate that each eighth grader must
meet the minimal standard in both reading and math to be promoted to the ninth grade, sought
both information and direction for the eighth grade curriculum design for “catching kids up” in
44
math. Concomitantly, within the achievement context, the outcomes of the subgroups can
eclipse the success of the majority. For example, if the majority group exceeds the standards and
the subgroups fail to meet the standards, this could preclude the school from meeting AYP.
The NCLB math challenge was that each school achieves “math excellence” (U.S.
Department of Education, 2006). What is inferred is that the goal is quintessential for both
international leadership and national security; it is most important that eighth graders achieve at a
level that will grant each student access to the high school math curriculum, which in turn will
grant students access to the universities’ curriculum. The NCLB advocates “scientifically based
methods with long-term records of success to teach math and measure student progress. There is
a need to “establish partnerships with universities to ensure that knowledgeable teachers deliver
the best instruction in their field” (U.S. Department of Education, 2006, n.p.). As the U.S.
Department of Education advocates a highly qualified teacher for each child, then it seems quite
equitable for schools to provide more than one teacher to help bridge academic gaps for LD and
LA students.
The presupposition and underlying implication of NCLB is that each child can learn
(Linn et al., 2002). Researchers have assumed that learning is measurable. Because this
implication of NCLB is measurable through criteria predetermined by the state of Georgia, if a
student fails to learn, academic intervention is quintessential. Best practices have suggested
academic intervention for remediation (Stonehill et al., 2011). Academic intervention is possible
because of the theory of multiple intelligences (Gardner, 1999); consequentially, appropriate
differentiation (Tomlinson, 1995a) best accommodates each child’s intelligence as well as
learning strategies. Best practices for remediation suggest that time on task and practice making
permanent be provided (Nomi & Allensworth, 2009). As time is most important, the
45
intervention(s) must be both efficient and effective (Gersten et al., 2009; Nomi & Allensworth,
2011). Thus, when a teacher discovers that a student did not learn from his/her teaching
methods, thereby causing the student to miss a significant amount of learning, it becomes
expedient to discover how to facilitate learning in a manner in which the student learns best and
to correlate the pedagogy with the student’s epistemology, including, but not limited to, content,
process, and assessment (Nomi, 2012; Nomi & Allensworth, 2013; Tomlinson, 1995b; 1999).
A pedagogy that includes the differentiation of the content, process, and assessment to
accommodate high student interest and how a student learns is something that a teacher can
control—teacher efficacy; however, more time is often needed to facilitate the bridging of
academic gaps (Goddard et al., 2004; Gersten et al., 2009). Thus, if given enough time to
intervene and if the intervention is individualized, the learning is impacted because of teacher
efficacy (Guskey & Passaro, 1998; Ross, 1994). Subsequently, it could substantially impact a
student’s summative learning outcome when aligned with the curriculum-based standards as
measured by the CRCT (Silva, 2007; Siwatu et al., 2009; Siwatu & Starker, 2014; Tucker et al.,
2005; Vaughan, 2002).
When instruction matches student needs, significant learning occurs (Wiles et al., 2006).
Neal and Schanzenbach (2010) examine some of the impact the Chicago Public Schools high
stakes testing had on the learning. Subsequently, there was a shift in pedagogy addressing
certain student needs in both math and reading. Subsequently, there was an increase “in the
middle of the achievement distribution but not among the least academically advantaged
students;” they suggest “that changes in proficiency requirements induce teachers to shift more
attention to students who are near the current proficiency standard” (p. 263).
46
Piaget’s cognitive development theory suggests that all underlying approaches must
consider that when instruction matches the student’s needs, significant learning often occurs
(Atherton, 2013). Not only should the learning be individualized, but the teacher must
frequently check for understanding, allowing him/her to identify both academic progress and/or
gaps (Marzano, 2003; Posner & Rudnitsky, 2006). Success in learning could suggest
empowerment, if done appropriately. As the student learning is scaffolded, the student may
develop a sense of empowerment through self-efficacy (Wehmeyer et al., 2012); however,
teachers need the appropriate time and context to accomplish both the appropriate education and
self-efficacy (Gamble et al., 2012).
Time on Task
Students who are low performing often need more time than allotted for the learning and
mastering of the content and skills (Cortes et al., 2015; Farbman, Christie, Davis, Griffith, &
Zinth, 2011; Patall, Cooper, & Allen, 2010); however, researchers have suggested that how and
when that time is allotted could have negative and positive effects. Traditionally, the time most
utilized has been either summer school or retention. In fact,
retained students scored 0.19 to 0.31 standard deviations below comparable
students who had not been retained. Moreover, a variety of studies have found
that retention is associated with an increased likelihood of dropping out (E.M.
Shulz et al., 1986; Russell W. Rumberger, 1987; James B. Grissom and Lorrie A.
Shepard, 1989; Michelle Fine, 1991; Melissa Roderick, 1994). Several more
recent studies have found moderate, positive effects of retention (Nancy L.
Karweit 1991; Louisa H. Pierson and James P. Connell 1992; Karl L. Alexander,
Doris R. Entwisle, and Susan L. Dauber 1995; A. Gary Dworkin, Jon Lorence,
47
Laurence A. Toenjes, and N. Hill Antwanette 1999; Eric R. Eide and Mark H.
Showalter 2001). (Jacob & Lefgren, 2009, p. 2)
Jacob and Lefgren (2009) argued that the retention of children and its future impact
seems to be relative. Xia and Kirby (2009) considered the impact of New York’s test retention
policy on fifth graders longitudinally from 2006-2009. Their review of 91 studies suggests that
the negatives outweigh the positives. Winters and Greene (2012) considered the impact of
Florida’s test retention policy of third graders during their following five years. The third
graders were required to go to summer school and receive a high-quality teacher for the next
year. “Exposure to these interventions has a statistically significant and substantial positive
effect on student achievement in math, reading, and science;” however, “the effect of the
treatment dissipates over time” (p. 305). Previously, Jacob and Lefgren (2004) found “no
consistent differences in the performance of retained versus promoted students in the short-run”
(Jacob & Lefgren, 2009, p. 2). Rather than exploring only the short-term academic focus, they
considered “the direct academic consequences of summer school and grade retention for those
students who fail to meet the promotional standards” (Jacob & Lefgren, 2009, p. 2). They argued
that if the first retention is in the sixth grade, then the dropout rate seems minimal to none;
conversely, there seems to be a negative correlation of retention in elementary school with eighth
graders’ self-efficacy. In fact, “retaining low-achieving eighth grade students in elementary
school substantially increases the probability that these students will drop out of high school” (p.
4).
As previously noted, research suggests that the differentiation of the content, process, and
assessment to accommodate high student interest and how a student learns is something that a
teacher can control—teacher efficacy; however, sometimes students need more time. Both
48
students and teachers need more time to facilitate the bridging of the academic gaps (Goddard et
al., 2004); however, the intervention must be individualized for effectiveness (Guskey &
Passaro, 1998; Ross, 1994). Subsequently, student assessments can be impacted if given enough
time and if each student is scaffolded through the best practices (Phelan et al., 2011).
Powell et al. (2013) suggested interventions for students with math difficulties (MD); this
includes students who have been officially identified with a disability and those who manifest
math difficulties. Teacher recommendation is operative for identifying students with MD. The
difficulties experienced by students with MD are primarily with foundational concepts. In
addition, many students struggle with one-to-one correspondence, language comprehension,
reading difficulties, and visual spatial limitations, even after they would be expected to perform
beyond these concepts to meet grade-level standards (Powell et al., 2013, p. 38).
When a student is at risk of failing, schools are to implement interventions (Jenkins,
Schiller, Blackorby, Thayer, & Tilly, 2013; Silva, 2007). Relative responses to interventions
(RTI) could substantially impact a student’s summative learning outcome when aligned with the
curriculum-based standards as measured by the CRCT (Silva, 2007; Siwatu, et al., 2009; Siwatu
& Starker, 2014; Tucker et al., 2005; Vaughan, 2002). Although research has shown that there
are numerous methodologies which may be used to catch students up, retention is still a
consideration by some (Brown, 2007; Brown et al., 2011; Smith & Bell, 2014).
Retention has mixed reviews in the literature (Orfield, Losen, Wald, & Swanson, 2004),
as some have purported an opportunity to be proactive before retention is required. A student at
risk of failing can be identified and RTI implemented (Burns & Gibbons, 2012). RTI suggests
that a student at risk of failing needs interventions that could include both more time on task and
individualized learning (Nomi & Allensworth, 2011). In fact, if identified early enough, students
49
at risk can be offered specific interventions to support them in the learning process. Some
research has suggested that a remedial math class is an acceptable RTI that teachers can
implement to meet the student’s need through “organization, affiliation, and product-focus
lessons” (Adams, 2011, p. 75; Bottge et al., 2004; Flores & Kaylor, 2007).
Summer school offers an opportunity to remediate students who have failed to meet
standards during the previous school year (Stone et al., 2005). Proponents of summer school
believe that schools should remediate. If remediation is deemed a necessity, it then becomes a
matter of how, when, and for how long (Foegen et al., 2007; Nomi & Allensworth, 2011; Redd et
al., 2012). There is a need in the literature to answer the above questions. In addition, both more
time on task and the practice of repetition making permanent are two presuppositions that need
to be considered. The lack of research suggests a need for this study and subsequent studies to
better determine an appropriate amount of time and the necessary best practices to address
meeting the needs of compensatory education for children at risk of failing (Cortes et al. 2015;
Foegen et al., 2007; Nomi & Allensworth, 2011).
While more time on task and practice making permanent are past approaches for
addressing remediation, some suggests more differentiation within that remedial time frame
(Kommer, 2006; Sax, 2006; Valero, 2012). Since a student did not learn from the initial learning
process, then the student needs to be taught in a different way (Brown et al., 2011; Smith & Bell,
2014). Some have suggested differentiation for even gender peculiarity (Ai, 2002; Carr &
Alexeev, 2011; Ganley et al., 2013; Lindberg, Hyde, Petersen, & Linn, 2010). Although gender
distinctiveness is relevant to some studies, it seems to be a moot point to others (Din, Song, &
Richardson, 2006; Forgasz & Rivera, 2012; Kane & Mertz, 2012; Lukas & Beresford, 2010).
50
The report of the National Mathematics Advisory Panel (2008) suggested that formative
assessment can be both informative and directional; both informing a teacher’s pedagogy and
directing the scaffolding process. Studies have shown that students need more individualized
pedagogy; therefore, “teachers’ regular use of formative assessments improves their students’
learning, especially if teachers have additional guidance on using the assessment results to design
and individualize instruction” (National Mathematics Advisory Panel, 2008, p. 47). In fact, “For
struggling students, frequent (e.g., weekly or biweekly) use of these assessments appears
optimal, so that instruction can be adapted based on student progress” (National Mathematics
Advisory Panel, 2008, p. 47). Although the research is not conclusive on the “specific tools and
strategies,” the panel suggestions include tutoring, computer assistance, and “a professional
(teacher, mathematics specialist, trained paraprofessional)” (National Mathematics Advisory
Panel, 2008, p. 47). Results of formative assessment “can lead to increased precision in how best
to use instructional time” and can “assist teachers in identifying specific instructional needs”
(National Mathematics Advisory Panel, 2008, p. 48).
Thus, the panel's conclusion of efficient and effective means of helping “students with
learning disabilities (LD) as well as low-achieving (LA) students” (National Mathematics
Advisory Panel, 2008, p. 48) is quite suggestive for this research. The underlying strategy for
both LD and LA is to receive, on a regular basis, some explicit systematic instruction that
includes opportunities for students to ask and answer questions and think aloud about the
decisions they make while solving problems (Schellings & Broekkamp, 2011). This type of
instruction can be incorporated without compromising the mathematics instruction that every
student receives (Rosenzweig et al., 2011). However, it does seem essential for building
proficiency in both computing and translating word problems into appropriate mathematical
51
equations and solutions (Krawec et al., 2012). A specific proportion of this time should be
dedicated to ensuring that students possess the foundational skills and conceptual knowledge
necessary for understanding the mathematical concepts they are learning at their grade level
(National Mathematics Advisory Panel, 2008, pp. 48-49).
Testing Format
High stakes testing in the state of Georgia utilizing the annual CRCT has had students
respond in multiple choice format. However, the proposed new testing suggests that the
assessment will include constructed-response format. Conversely, as early as Linn et al. (1991),
disparity and test inequalities between the two testing formats of multiple choice answers (MCA)
and the open end questions, the precursor to the constructed response (CR), have been argued
(Katz, Bennett, & Berger, 2000).
Research is mixed on the relevance of test reliability being impacted by the two distinct
testing formats. Since 1958, Powell (2012) has considered the impact of incorrect answers with
multiple choice tests and has identified three concerns—selective reasoning, Piaget’s reasoning
stages, and change in answers after repeated opportunity—all of which are consistent with
development theory in reasoning. Powell’s research, along with that of others, suggests that the
understanding of the question is more operative than the correct answer. For example, rote
memory facilitating a correct answer lacks consistency in validating acquisition of knowledge.
Thus, testing may not communicate to the teacher effectiveness or depth of knowledge (Iorio &
Adler, 2013). Instead, Powell suggested using a selection-pattern-analysis to evaluate the
student’s understanding.
Haladyna (1999) suggested that test reliability can be threatened through constructed
52
response; however, the multiple choice format offers more consistency in establishing reliability.
Bridgeman (2005) and Lukhele, Thissen, and Wainer (1994) argued that there is no relative
difference. On the other hand, there are those who have argued, based on their research, that
format does have relevance. Both the high school research of Bennett, Rock, and Wang (1991)
and Garner and Engelhard (1999) suggested that format has relevance. In fact, their research
found that, for whatever reason, high school students correctly answered a greater amount of
multiple choice questions than constructed responses.
Both multiple choice answers and constructed responses have distinct advantages and
disadvantages. While the state of Georgia suggests that a depth of knowledge must be
demonstrated through the CR, some would argue that the MCA demonstrates depth of
knowledge by being able to recognize the best answer. In fact, as far as recognizing the best
answer, the MCA outweighed the CR relative to the complexity of the answer. Caygill and Eley
(2001) established that a student’s proficiency to recognize the correct answer outweighed the
ability to construct an answer at the same level. Also, Ku (2009) suggested that the two types of
testing formats are administratively quite suggestive. While Bennett et al.’s (1991) research
argued for the reliability of the MCA, Lukhele et al.’s (1994) research argued for both the
efficiency and effectiveness of the MCA.
Although some research has suggested that the multiple choice format is much more
efficient, there are others who have challenged its veracity in all situations relative to depth of
knowledge. Some have argued that the MCA format benefits readers with limited proficiency,
minorities, females, and those with low socioeconomic status (Bloom’s Taxonomy,1956; Garner
& Engelhard, 1999; Griffin & Nix, 1991; Hambleton & Murphy, 1992) in demonstrating what
they can recognize.
53
Although the research is divided on the relevance of multiple choice format versus
constructed response, the fact remains that Georgia state assessments will contain both.
Subsequently, educators must respond. Tankersley (2007) suggested that because the rules for
testing are changing, teachers must prepare students by utilizing teaching strategies for
constructive answers; “educators must know and understand the ‘rules’ by which the score is
kept” (Tankersley, 2007, p. 3). Learning must include ways “that allow students to build the
skills, learn self-assessment, and provide a supportive and meaningful environment”
(Tankersley, 2007, p. 3). In particular, testing accommodations for students with special needs
will be impacted as students move from recognizing the depth of knowledge answer to
constructing the same depth of knowledge answer.
Students with disabilities are frequently granted accommodations for high-stakes
standardized tests to provide them an opportunity to demonstrate their academic
knowledge without interference from their disability. One type of possible
accommodation, test response format, concerns whether students respond in multiple-
choice or constructed-response format. (Powell, 2012, p. 3)
An experimental study was conducted to assess the performance differences of third
grade students, identified as having mathematical difficulties, on a test of mathematics problem
solving as a function of response format. It was found that “students responding in the multiple
choice format had a significant advantage over students answering in the constructed response
format” (Powell, 2012, p. 3). While some students may have been impacted more than others by
the change in the testing format, this study sought to study the impact of the math-related testing
format on low performers. As such, the literature suggests that the multiple choice format has
implications for low achievers, particularly students who qualify for a testing accommodation.
54
As suggested by Fuchs and Fuchs (2001), an accommodation if appropriate,
provides students with disabilities with a differential boost over students without
disabilities. With a differential boost, students with disabilities benefit
substantially more from the accommodation than students without disabilities.
Determining whether this accommodation of multiple-choice format provides a
differential boost for students with disabilities requires additional research, in
which the performance of students with and without disabilities is directly
compared on a constructed-response and multiple-choice version of the same
mathematics test. (Powell, 2012, p. 8)
Educators have no choice in the matter of test format on the state level. Therefore,
whether the state of Georgia uses the multiple choice answer or constructed-response format, a
student must have a core of knowledge to answer correctly. In addition, it is essential that
students become familiar with the new testing format. What is most disconcerting is that a
student with cognitive disabilities such as long-term retrieval and executive functioning may be
at a distinct disadvantage. Strategies that consider both the retention of knowledge (Roediger,
Agarwal, Kang, & Marsh, 2010) and the recognition of that same knowledge (Johnson, Hedner,
& Olsson, 2012) are both indicative of the educators’ mandate to give each child a FAPE. Not
only does the new testing format suggest new challenges for students who might be
disadvantaged, but the results have implications for educators in particular. With the new
Teacher Keys Effective System (TKES), the stakes for educators are being raised in the realm of
accountability, especially if there is not relative growth in testing scores. However, new
strategies will be implemented as educators seek to give each student a FAPE (Chan, 2010;
Tankersley, 2007).
55
Summary
Compensatory education is possible. As schools have attempted to address interventions
for students at risk and find better ways to include students with special needs in the general
classroom, efficiency and effectiveness have become paramount (Cosier et al., 2013). As every
child can learn, educators must discover each student’s epistemology and scaffold the learning
through formative assessment, bridging each gap along the journey toward proficiency in the
assigned task. Appropriate differentiated interventions can make a difference (Gersten et al.,
2009; Nomi & Allensworth, 2011). In the shadow of a plethora of interventions available, this
research considered the following question: Was attendance in a nine-week MC class an
effective means for improving math scores of those who were identified as low-achieving eighth
grade students at a Georgia middle school?
This quasi-experiment did not allow for randomization; rather, the students were assigned
based upon the academic need. Sometimes randomization precludes the neediest from receiving
the much needed treatment. Ethically, when an individual or group is most needy, it is
acceptable to systematically select those who are most needy; perhaps this is a more equitable
and intentional systemic process that is quite suggestive and predictable—removing any
uncertainty of the criteria used for determining the ones receiving the treatment (Berk & Rauma,
1983; Rossi, Lipsey, & Freeman, 2004; Trochim, 1984).
There were two key assumptions regarding this research. First, if students should show
significant improvement from the AIMSweb pretest to the posttest, the pretest, treatment, and
posttest should be most informative and quite suggestive. Second, since the eighth grade
curriculum is aligned to the state of Georgia’s summative assessment, the CRCT, these results
will help to realign and redirect the instruction for future curriculum and instruction. If effective,
56
it may prove to be an efficient approach in addressing the low achievers; concomitantly, it may
be directional for each student’s eligibility for promotion to the ninth grade.
57
CHAPTER THREE: METHODS
Research Design
This quantitative study used a quasi-experiment nonequivalent control group design; a
pretest and posttest were used with the AIMSweb tests. The AIMSweb tests served as both the
pretest and posttest. Groups were preassigned based on a cutoff score or a math teacher’s
recommendation. The AIMSweb data were used to consider growth, if any, that the math
connection class might have contributed.
Presently, the trend towards the quasi-experimental design seems to be increasing due to
necessities created by preassigned groups (Aiken, West, Schwalm, Carroll, & Hsuing 1998;
Berk, Barnes, Ahlman, & Kurt, 2010; Shadish, Galindo, Wong, Steiner, & Cook, 2011). In fact,
Imbens and Lemieux (2008) have identified that since the late 1990s, the field of economics has
experienced an increase in quasi-experimental designs” (Angrist & Lavy, 1999; Black, 1999;
Card et al., 2006; Chay & Greenstone, 2005; Chay et al., 2005; DiNardo & Lee, 2004; Lee,
2007; Van Der Klaauw, 2002)” (p. 618). The independent variable was the extra nine-week
math class—those who participated and those who did not. The dependent variable was the
AIMSweb posttest score for each student. The covariate was the AIMSweb pretest scores for
each student.
Research Question
The research question for this study was:
RQ1: While using the AIMSweb pretest scores as a control variable for previous math
achievement, will at-risk eighth grade students who attend an intensive nine-week compensatory
math class have statistically significant different mean scores as measured by the AIMSweb
posttest when compared to students who do not receive the compensatory math instruction?
58
Null Hypothesis
The null hypothesis for this study was:
H01: While using the AIMSweb pretest scores as a control variable, at-risk eighth grade
students who attend an intensive nine-week math connection class will not have statistically
significant different mean scores as measured by the AIMSweb posttest when compared to
students who do not receive the compensatory math instruction.
Participants and Setting
The participants for this study consisted of a convenience sample of 189 eighth graders.
Although this Georgia suburban, Title I middle school houses sixth, seventh, and eighth graders,
only the eighth graders were selected for participation. The limited focus had both economic and
equitable concerns; an eighth grader had to pass the math CRCT as a requirement to be promoted
to the high school. To be able to access higher mathematics courses, it was imperative that the
eighth grader be well prepared academically for the challenges of high school freshmen courses.
This study was limited to just one school. Again, this was a Georgia suburban middle
school in suburban Atlanta with 688 total students. The student population was ethnically
diverse (Black = 53%, White = 26%, Multi-Racial = 9%, Asian = 7%, Other = 5%). The
percentage of students eligible for free lunch was 32% and those eligible for reduced lunch was
10%. The number of participants sampled was 189, which exceeded the required minimum for a
medium effect size (Cook & Campbell, 1979; Olejnik & Algina 2000; Shadish, Cook, &
Campbell, 2002). According to Gall, Gall, and Borg (2007), 96 students is the required
minimum for a medium effect size with “statistical power of .7 at the .05 alpha level” (p. 145).
The treatment group was based upon one of two criteria: to be in the treatment group, (a)
a student had to have a score of 820 or below on the previous year’s seventh grade CRCT state’s
59
annual math assessment, or (b) a student had to be referred to the group by the student’s eighth
grade math teacher because the student was at risk of failing. The control group was all other
students who took the seventh grade math state standardized CRCT tests and were assigned to a
non-math connection class.
The current study administered the AIMSweb test to 189 respondents who were included
in this study. The participants were almost equally split female (51.9%) and male (48.1%). Only
13.2% of participants were classified as gifted overall, with the treatment group having 10.4%
gifted and the control group having 14.8% gifted. The majority of respondents were African
American (55%), followed by Whites (30.2%), and Asians (6.9%). Table 3-1 contains the
frequencies for gender, gifted, and ethnicity by control and treatment groups.
Table 3.1
Frequencies: Demographics
Control Treatment Total (189)
N % N % N %
Gender
Female 62 50.8% 36 53.7% 98 51.9%
Male 60 49.2% 31 46.3% 91 48.1%
Gifted
No 104 85.2% 60 89.6% 164 86.8%
Yes 18 14.8% 7 10.4% 25 13.2%
Ethnicity
Asian 10 8.2% 3 4.5% 13 6.9%
African American 62 50.8% 42 62.7% 104 55.0%
Hispanic 3 2.5% 2 3.0% 5 2.6%
Indian 1 0.8% 0 0.0% 1 0.5%
White 41 33.6% 16 23.9% 57 30.2%
Other 5 4.1% 4 6.0% 9 4.8%
All eighth graders were given the opportunity to participate; however, logistically they
were required to be available and willing to participate on the days of both the pre and posttest.
60
Also, these middle school MC classes were preassigned, but students could request a change in
schedule. Students were placed into the additional MC class based upon two factors. One factor
for consideration was the score from the 2010 mathematics section of the CRCT. Those students
scoring at or below 820 (the 25th percentile) were candidates for the MC class. A secondary
factor for the class was teacher recommendation. Teacher recommendations came from the
current 2010-2011 school year mathematics general education classroom instructors who were
familiar with the students’ abilities and whether the students were at risk of failing the CRCT for
the current testing year. The assistant principal then placed those students who qualified for the
additional class into the program; therefore, the groups were nonrandomized.
Instrumentation
The AIMSweb (nonmultiple choice format) tests measure math performance. The
AIMSweb has been used by many researchers (Graney, Missall, Martinez, & Bergstrom, 2009;
Lembke, Hampton, & Beyers, 2012; Riccomini & Witzel, 2009; Shapiro & Gebhardt, 2012).
Both tests were administered by trained certified teachers in accordance with secured testing
procedures and regulations utilizing paper and pencil on separate designated days.
Confidentiality was maintained by assigning each student a number.
AIMSweb
The AIMSweb (AIMSweb, 2009) is a norm-referenced assessment that adheres to the
National Council of Teachers of Mathematics Principles and Standards (NCTM, 2006) and the
Stanford Achievement Test, Tenth Edition (Stanford 10). It is a web based solution that provides
real-time information to teachers and schools for students in grades 2-8.
The purpose of the AIMSweb was to monitor and report student progress in math and
identify at risk students early (AIMSweb, 2009). The school utilizes both the AIMSweb and the
61
CRCT, a criterion based assessment tool, to measure curriculum specific information established
by the Georgia Department of Education (GaDOE). So, one test provides feedback on GaDOE
specific standards, and the AIMSweb provides feedback on performance related to national
norms.
The AIMSweb was administered. There were a total of 30 questions both in the pretest
and posttest. The 10 minute time frame tested both computation and processing speed associated
with the students’ grade level. Computation skills were assessed. For example, mathematic
computation included column addition, basic facts, decimals, reducing, and exponents. Scoring
was based upon recording the number correct. Both pretest and posttest scores were recorded,
compared, and analyzed through web based software.
Experienced item writers with expertise in mathematics curriculum created
approximately 11,200 items in accordance with grade-level and domain-specific criteria. Three
pilot studies were conducted to evaluate the items and finalize probe design prior to the field test.
A national field test was conducted at each grade. Forty-four clones of the anchor probe were
constructed, consisting of items parallel to the anchor-probe items. Each clone had the same
sequence of item types as the anchor probe. All probes were administered to a national field test
sample of 6,550 students in the spring of 2009.
To assess the construct validity of the AIMSweb math test, AIMSweb scores were
correlated with the North Carolina End of Grade math test and the Illinois Standards
Achievements test. The correlation coefficient of the AIMSweb math scores for all covered
grades (2-8) ranged from .60 to low .70. Based on Cohen’s effect size standards, correlation
coefficients of .5 or above are considered strong (Cohen, 1988). Additionally, Chronbach’s
62
alpha reliability scores ranged from .80 to .88, indicating acceptable reliability (Field, 2012;
Pallant, 2013).
Procedures
All approvals were granted, school and then IRB, for the study (See Appendixes A for
school approval and B for IRB approval.). Initially, the MC class was for students who scored
an 820 or below on the previous year’s CRCT in math; however, some teachers were concerned
that there were other students who were at risk of failing the 2011 CRCT. Because there was a
lack of content correlation between the seventh and eighth grade CRCT, the administrator
followed the teacher’s recommendation by extending the class to those recommended by the
student’s 2010-2011 math teacher, even if they had scored above 820 on the previous year's math
CRCT.
The county in which the middle school is located uses the AIMSweb program to track the
progress of its students in the areas of reading, writing, and mathematics. The middle school
eighth grade math teachers use the Georgia GPS Edition COACH Standards-Based Instruction
Math Grade 8 book (2008), along with the scope and sequence chart provided by the county, as a
guide for classroom instruction. The COACH (2008) book contains a pretest and a posttest as
well as mini-lessons and practice questions to aid classroom instructors in preparing students for
the CRCT given in the spring of each school year. The eighth graders were given the AIMSweb
pretest during the first week of the 2010-2011 school year; the posttest was administered in the
spring before the CRCT, but after the completion of the MC. These pretest scores, in turn, were
used to provide this researcher with a baseline for the study.
Since the middle school utilizes both the AIMSweb and the COACH (2008) curriculum,
teachers had been trained in both the teaching and assessment aspects. Trained teachers
63
administered, scored, and recorded both the AIMSweb pretest and posttest; scores were
confidentially maintained on a private computer. Once the post test scores were matched to the
pretest, the computer software generated a student ID number for each student, demographic
data, and then all names were deleted. This was the same procedure for the CRCT test scores.
Also, the students who were recommended by their present year teachers to participate in the MC
class were identified. Three teachers checked all the data for accuracy.
Sixty-seven students who were assigned to the treatment group, those receiving the nine-
week Math Connection (MC) class, were given additional math instruction for 55 minutes a day
for 45 consecutive school days. Connection classes are any classes that are non-academic which
serve as electives. Six special connection classes were created for the students in the treatment
group. These 67 students were divided into six separate classes utilizing certified math teachers
and implementation of the COACH (2008) curriculum, an overview of the standards tested by
the CRCT. The students were assigned based upon perceived need. The coursework in the
treatment connection class was guided by a COACH (2008) workbook. The students and
instructor worked through each page of the COACH (2008) workbook for 45 consecutive days,
at which time they completed the entire workbook. The COACH (2008) workbook was modeled
after the CRCT.
Data Analysis
To answer the research question, an ANCOVA was used. The analysis of covariance
(ANCOVA) examines differences between two or more groups on a continuous variable, while
controlling for the effects of one or more variables (Ary et al., 2010; Tabachnick & Fidell, 2012).
In this analysis, AIMSweb pretest scores were the covariate, AIMSweb posttest scores were the
dependent variable as distinguished in the two groups of eighth graders (general vs. at-risk).
64
Before an analysis of covariance (ANCOVA) was conducted, several preliminary tests were
completed to determine if the assumptions needed to perform an ANCOVA were met. These
assumptions include normality, homogeneity of variance, linearity, and homogeneity of
regression slopes (Edmonds & Kennedy, 2013; Field, 2012; Tabachnick & Fidell, 2012).
Normality was tested using the Kolmogorov-Smirnov test, where a p value of less than .05
indicates non-normality. Levene’s test of homogeneity of variance was used to assess
homogeneity of variance, where a p value of less than .05 indicates a violation in the assumption
of homogeneity of variance. The assumption of linearity was checked by generating a scatterplot
between pretest and posttest scores for control and treatment groups. If the distribution of scores
for both groups is linear, then the assumption of linearity holds. If the distribution of scores is
curvilinear for at least one of the two groups, then the assumption of linearity is violated.
Finally, the assumption of homogeneity of regression slopes was assessed to evaluate if there
was an interaction between the covariate and the dependent variable. If the interaction term in
the ANCOVA is significant, meaning a p value of less than .05, then there is a violation in the
assumption of regression slopes.
By using an analysis of covariance (ANCOVA), the two groups’ differences in student
math performance on the AIMSweb pre and posttests were controlled for and better understood.
The ANCOVA allowed for the distinction, if any, of the impact of the MC class (treatment
group) in relation to the group that did not receive the MC class (control group) (Ary, Jacobs,
Razavieh, & Sorensen, 2010).
65
CHAPTER FOUR: FINDINGS
Research Question
The research question for this study was as follows:
RQ1: While using the AIMSweb pretest scores as a control variable for previous math
achievement, will at-risk eighth grade students who attend an intensive nine-week math
connection class have statistically significant different mean scores as measured by the
AIMSweb posttest when compared to students who do not receive the compensatory math
instruction?
Null Hypothesis
The null hypothesis for this study was:
H01: While using the AIMSweb pretest scores as a control variable, at-risk eighth grade
students who attend an intensive nine-week math connection class will not have statistically
significant different mean scores as measured by the AIMSweb posttest when compared to
students who do not receive the compensatory math instruction.
Descriptive Statistics
There were a total of 189 respondents who took part in this study. The descriptive
statistics include the variables measured by the AIMSweb pretest scores and the AIMSweb
posttest scores. These descriptors include the range, mean, median, and standard deviation. The
descriptive statistics for the AIMSweb pretest and post scores are located in Table 4.1.
66
Table 4.1
Descriptive Statistics for AIMS Web Pretest, Posttest Test Scores
Test Range M Median SD
Control
AIMS Web Pretest 3-25 10.54 10.00 3.50
AIMS Web Posttest 4-28 11.98 11.00 4.55
Treatment
AIMS Web Pretest 2-14 7.48 7.00 2.81
AIMS Web Posttest 4-19 9.12 8.00 3.52
Total
AIMS Web Pretest 2-25 9.46 9 3.58
AIMS Web Posttest 4-28 10.96 10 4.42
Results
Data Screening
Data screening was conducted on each group’s AIMSweb pretest and posttest scores
regarding data inconsistencies and extreme outliers. Frequency distributions were generated for
each of the two variables across all respondents and scanned for inconsistencies. No data errors
or inconsistencies were identified. Additionally, the box and whisker plots revealed no extreme
outliers. Outliers were observed, however, but the ANCOVA is a robust test, meaning mild
violations in skewness, normality, and equal variances will still yield p values within ± .02 of the
true p value (Boneau, 1960; Posten, 1984; Schmider et. al., 2010). See Figure 4.1 for the box and
whisker plots for the AIMSweb pretest and posttest scores by group.
67
Figure 4.1: Box and whisker plots of AIMSweb pretest and posttest scores by control and
treatment groups reveals no extreme outliers.
Assumptions
An Analysis of Covariance (ANCOVA) was used to test for significant differences in
AIMSweb posttest scores between control and treatment groups when controlling for AIMSweb
pretest scores. The ANCOVA required the assumptions of normality, linearity, bivariate normal
distribution, the assumption of equal variances, and the assumption of homogeneity of regression
slopes. Results of the Kolmogorov-Smirnov test indicated that there was a violation in normality
for both the AIMSweb pretest, KS(189) = .108, p < .001 and the AIMSweb posttest, KS(189) =
.126, p < .001. See Table 4.2. Results of Levene’s test of homogeneity of variance indicated
that there was no violation in homogeneity of variance for the AIMSweb pretest, F(1, 187) =
68
3.023, p = .084 or the AIMSweb posttest, F(1, 187) = 3.032, p = .083. See Table 4.3. To assess
linearity and bivariate normal distribution, a series of scatterplots were generated between the
AIMSweb pretest and posttest variables for each group. The results revealed that the majority of
the plots for the control group formed the desired cigar shape for the AIMSweb pretest, but for
the treatment group, there was slight heteroscedasticity as the pretest and posttest values
increased. See Figures 4.2 and 4.3. The assumption of homogeneity of regression slopes was
assessed to evaluate if there was an interaction between the covariate and the dependent variable.
If the interaction term in the ANCOVA is significant, meaning a p value of less than .05, then
there is a violation in the assumption of regression slopes. Results indicated that the pretest
AIMSweb group interaction term was not significant, F(1, 185) = .013, p = .911. Therefore,
there was no violation in the assumption of homogeneity of regression slopes. See Table 4.4.
Given the results of the tests of assumptions, it was deemed appropriate to perform the
ANCOVA as there was no violation in the assumption of regression slopes, and the ANCOVA is
robust to violations of normality, skewness and equal variances (Boneau, 1960; Posten, 1984;
Schmider et. al., 2010).
Table 4.2
Kolmogorov-Smirnov Test of Normality
Statistic Df p
AW PRE .108 189 .000
AW POST .126 189 .000
69
Table 4.3
Levene’s Test of Homogeneity of Variance
Levene
Statistic df1 df2 p
AW PRE Based on Mean 3.023 1 187 .084
AW POST Based on Mean 3.032 1 187 .083
Figure 4.2: Scatterplots of AIMSweb pretest and posttest scores from the control group has the
desired cigar shape.
70
Figure 4.3: Scatterplots of the AIMSweb pretest and posttest scores for the treatment group
display slight heteroscedasticity at the higher pretest and posttest score levels.
Table 4.4
ANCOVA Table: Test of Homogeneity of Slopes
Source SS df MS F p
Corrected Model 1554.452a 3 518.151 45.210 .000
Intercept 243.125 1 243.125 21.213 .000
Treatment 2.002 1 2.002 .175 .676
AWPRE 936.964 1 936.964 81.752 .000
Treatment * AWPRE .145 1 .145 .013 .911
Error 2120.289 185 11.461
Total 26390.000 189
Corrected Total 3674.741 188
71
Null Hypothesis One
H01: While using the AIMSweb pretest scores as a control variable, at-risk eighth grade
students who attend an intensive nine-week math connection class will not have statistically
significant different mean scores as measured by the AIMSweb posttest when compared to
students who do not receive the compensatory math instruction.
A one-way analysis of covariance (ANCOVA) was conducted to determine if there was a
statistically significant difference in mean scores on the AIMSweb post math tests between the
treatment and control groups. The independent variable was the extra nine-week math class —
those who participated and those who did not. The dependent variable was the AIMSweb
posttest score for each student. The covariate was the AIMSweb pretest score for each student.
Necessary assumptions were considered and analyzed. The results indicated that there was no
significant difference in adjusted mean scores between the control group (Madj = 11.13, SE = .32)
and the treatment group (Madj = 10.65 SE = .44) when controlling for pretest scores, F(1, 186) =
.730, p = .394. The eta squared effect size measure was η = .004, indicating .4% of the
variability in posttest scores was accounted for by treatment group. Based on Cohen’s (1988)
guidelines where .01 is a small effect, .06 a medium effect, and .14 a large effect, the effect was
small. See Tables 4.5 and 4.6.
Table 4.5
ANCOVA Adjusted Mean Scores for the Treatment Groups
Treatment N Madj SE 95% CI LL 95 CI UL
Control Group 122 11.13 .32 10.509 11.758
Treatment Group 67 10.65 .44 9.787 11.518
72
Table 4.6
ANCOVA Table: Assessing Difference in AIMSweb Posttest Scores
SS df MS F p Eta Squared
Corrected Model 1554.307 2 777.153 68.170 .000 .423
Intercept 273.636 1 273.636 24.003 .000 .114
AWPRE 1201.537 1 1201.537 105.396 .000 .362
Treatment 8.318 1 8.318 .730 .394 .004
Error 2120.434 186 11.400
Total 26390.000 189
Corrected Total 3674.741 188
73
CHAPTER FIVE: DISCUSSION, CONCLUSIONS, AND RECOMMENDATIONS
Discussion
The purpose of this quantitative quasi-experimental study was to see if an intensive nine-
week remedial math connection class (based upon the eighth grader’s seventh grade math CRCT
scores) can significantly increase scores. The dependent variable is the AIMSweb posttest scores
in math. The independent variable is where one group participates in the nine-week remedial
math training and the other group will not. Given the low proficiency in math among eighth
grade students, there is a need to intentionally offer equitable education to bridge the academic
gaps for eighth grade students at risk of failing (Jitendra, 2013; Malmgren et al., 2005; Munk et
al., 2010; Schrank & Wendling, 2009).
The research question asked if while using the AIMSweb pretest scores as a control
variable for previous math achievement, will at-risk eighth grade students who attend an
intensive nine-week MC class have statistically significant different mean scores as measured by
the AIMSweb posttest when compared to students who do not receive the compensatory math
instruction. The null hypothesis was that there was no statistically significant difference in
posttest scores, when controlling for pretest scores, between those in the nine-week MC class and
those who did not receive the nine-week MC class. An ANCOVA was conducted to answer the
research question and test the null hypothesis. The results of the ANCOVA indicated that there
was no statistically significant difference between eighth grade students who took the nine-week
math connection class and those eighth grade students who did not. Therefore, the null
hypothesis was not rejected.
74
Interpretation of Findings
The findings of the research question and the related null hypothesis showed that there
was no significant difference in AIMSweb posttest scores between eighth graders who were in a
remedial nine-week MC course and those who did not take the nine-week MC course. Although
the results were not significant, they were positive in that eighth grade students who had been
identified as needing math remediation had scored equally as well on the AIMSweb posttest as
eighth graders who did not need remediation. Therefore, the nine-week MC class was effective
at improving math performance of the remedial group from the pre-test to the post-test by one
point.
Findings in Context of the Literature
Researchers have suggested that more time spent on remediation tasks such as summer
school, retention, after school programs, and double math classes have impacted some learners
(Jacob & Lefgren, 2004, 2009; Matsudaira, 2007). Furthermore, Methe et al. (2012) looked at
the math skill remediation and analyzed the effect size through a meta-analyses of 47 effects in
11 studies (effect sizes ranged from 0.59 to 0.90). They found that the variables that appeared to
moderate the effects were student age, time spent in intervention, and intervention type. No
research has examined the effectiveness of an intensive nine-week remedial MC class to improve
math scores on the Georgia CRCT eighth grade math test. Thus, the findings of this study
extend the literature on time spent on remediation by concluding that the nine-week MC class,
which provided additional time spent on math remediation, indeed had a positive effect by one
point from the pretest to the posttest.
75
Findings in Context of the Theoretical Framework
The first theoretical framework used for this study was Piaget’s stage of cognitive
development, specifically, the concrete operational theory. Piaget’s concrete operational theory
suggests that cognitive development has matured in the learner to the point that the learner can
both learn math rules and then apply those rules to physically perceived objects (Atherton,
2013). The findings of this study revealed that eighth grade students in the nine-week MC class
improved their performance in math to be equal to that of their non-remediated peers. This aligns
with Piaget’s concrete operational theory since the students were able to learn math rules and
apply them to a physically perceived object in the form of test questions on a standardized math
exam. Based on Piaget’s theory, the students in the nine-week MC class demonstrated the
matured cognitive development necessary to learn.
A second theoretical framework used for this study was the constructivist theory.
Constructivists suggest that the learners can take information and connect it in such a way that
implementation is evident by what the learner is able to construct and communicate (Ertmer &
Newby, 2013). Constructivists use the social interaction of the teacher with the student to create
a learning environment that supports the student’s learning by implementing a social cognitive
process that is both suggestive and directional—modifying and adapting the learning to meet the
student’s most specific needs—which subsequently empowers the learner by creating and
perpetuating a sense of self-efficacy (Bandura, 1997; Marzano, 2003; Posner & Rudnitsky,
2006). Thus, both direct instruction and time on task are expedient for learning the math rules
and then moving those rules from short term memory to long term memory (Barbash, 2012;
Ertmer & Newby, 2013). The nine-week MC class was an interactive learning environment
between certified math teachers and students needing remedial math assistance. The guidance
76
from the math teachers was suggestive and directional in nature. The teachers did not give
students the answers to the workbook problems, but rather encouraged students to think about
the math problem in ways that were relevant for them. Based on the results of the study, the
students in the MC class were able to connect with information that was taught and then
communicate their knowledge in the desired manner via the AIMSweb posttest. The results,
therefore, support the constructivists theory of learning.
Conclusions
Based upon the previous discussion and the existing body of literature, it seems that some
students need more time on task to succeed in eighth grade math. Acceptance of failure cannot
be a consideration. Educators must find both effective and efficient pedagogy to educate each
student, FAPE.
Perhaps “we can overcome” failure through a collective efficacy that says “we can do it”
(Blankstein, 2013; Hattie, 2009, 2011). Collective efficacy seems to be essential in addressing
failure. The “concept of the ‘throw away’ students is itself discarded. Even the most abused and
troubled children self-correct as they mature” (Blankstein, 2013, p. 113). Learning obstacles do
exist: family opposition, language and culture distinctiveness, learning styles, more time on task
needed to become proficient, learning disabilities and other health impairments, and
socioeconomic status. “In high performing schools, these variables are addressed in a proactive
manner so they do not become barriers to the successful achievement of all students”
(Blankstein, 2010, p. 113).
Compensatory education is possible. As every child can learn, educators must discover
each student’s epistemology and scaffold the learning through formative assessment, bridging
each gap along the journey toward proficiency in the assigned task. Interventions can make a
77
difference (Gersten et al., 2009; Nomi & Allensworth, 2011). For example, those gaps are quite
challenging when students have learning disabilities; however, acceptance of those disabilities
and supporting them with an Individual Educational Plan (IEP) can be compensatory.
Concomitantly, when each student’s barrier to learning is addressed, success is inevitable.
Sometimes the “challenge is getting all staff members to believe” and to then “act on this
information in a sustained, concerted, systemic manner” (Blankstein, 2010, p. 113).
An IEP identifies and implements how a student can learn, even those who lack
motivation. The epistemology informs the team how to best support the learning of a student at
risk of failing. Student success is based on the student’s epistemology and the learning being
scaffolded in light of formative assessment and subsequent gaps being bridged until the student
can learn and implement the strategies without the support. Success is growth as measured by
the collected data assessing the individual goals.
Pedagogy must be informed by each student’s epistemology. Thus, nine weeks (55
minutes per day) may not be enough time on task for the remediation of some students at risk.
Early formative assessment can assist in a FAPE for each student; remediation for the most
“needy” is imperative to insure future access to the curriculum. If not, this inequitable access has
future socioeconomic consequences both on the individual and global level (i.e., global
competitiveness). Therefore, efficiency and effectiveness seem to be corollaries; however, the
questions of how much time is needed, and what are the research-based strategies that can
facilitate the individual learner must be assessed early to develop learner specific strategies.
While the schools move toward implementing a curriculum that is, or is similar to, the
Common Core, future researchers must consider the implications of the shift in both content and
assessment. Identification of inequities suggests the need for compensatory education that gives
78
access to a math curriculum (FAPE). In the wake of NCLB, educators still have the political
pressure to become both effective and time-efficient.
As schools move toward the new state assessments and with a limited amount of research
considering the impact of those attending a MC class as a measure upon student achievement
(Adams, 2011), new research will be considered. In relation to standardized testing, researchers
have suggested that content, processes, and assessments are quite directional for the remediation
process (Iorio & Adler, 2013; Powell, 2012). Tests like the AIMSweb are more challenging than
the CRCT as the answers are open ended for the AIMSweb, but multiple choice for the CRCT.
This open ended format could have an impact of test outcomes for certain students. Perhaps
many states moving toward the new testing format can find this research to be both informative
and directional for student achievement as measured by the new testing format, the new state
assessments. As FAPE is the goal, and budgets drive education, the question of effectiveness
and efficiency must propel future research.
Education is a public trust. Those charged with the fiduciary oversight must be held
accountable at all levels, especially using research-based praxis. Research-based praxis suggests
learning is possible. Pedagogy must transcend past failures and bridge academic gaps through
equitable education (Martinez & McGrath, 2014; Rickles, 2013). To ignore both the
epistemology and research-based praxis is to fail the learner and public trust.
Academic gaps suggest failure with socioeconomic implications (Goodman, 2012).
Sociologists Bourdieu and Passeron’s “theory of reproduction in education” has identified
“achievement gaps” that are actually socioeconomic “opportunity gaps” (Huddleston, 2014, p.
5). Lareau (2003) has suggested that these socioeconomic inequities have contributed to
academic gaps for the non-dominant groups. U.S. education has reproduced a middle class
79
model consistent with certain dominate socioeconomic values, resources, and skills that exist in
the local middle class families; teachers teach to the middle to perpetuate these educational
outcomes. Bourdieu and Passeron’s (1990) theory suggests that non-dominant students need
equitable opportunities to address the achievement (opportunity) gaps.
A child left behind infringes upon a nation's collective efficacy. Academic gaps in math
have socioeconomic implications (Domina, 2014). Just as reading facilitates the quality of one’s
life, so does math. Perhaps economic success, including jobs and global competition, is
contingent on one’s proficiency in math. Perhaps each gap bridged suggests better collaboration
as a nation, which, in turn, strengthens the nation’s global position.
Implications
Positive Social Change
Although the results were not significant, they were positive. The results demonstrated
that students who were identified early as needing math help were successfully remediated
within the same school year. One of the implications of the findings is that subject level
academic disparities in middle school may need not extend beyond a single school year. If
student weaknesses are identified in the beginning of the school year, the study shows that
effective remedial action in math among middle school students can be taken to erase the
performance deficit by the end of the same school year. If more academic institutions added
diagnostic formative assessments to their curriculum and supplied effective remedial assistance
to those who needed it, then, based on the research findings, this could significantly improve the
overall education of students in the United States.
80
Policy Makers, Administrators, and Teachers
The literature suggests that more time spent on math over and above what is allotted in
the classroom can be effective (Hall et al., 2011; Stone et al., 2005). However, the breadth of
effectiveness of after school math, summer school, retention for another year, and two math
classes has been debated (Jacob & Lefgren, 2004, 2009; Matsudaira, 2007; Roderick & Nagaoka,
2005). The results of this particular nine-week MC program imply that math remediation should
be further explored and considered over remediation approaches such as summer school, after
school math, and school retention for another year. The MC program needs to be replicated in
other schools, districts, and states to evaluate if the positive results are consistent across
populations. However, in this study no significant difference was found and the true
effectiveness of MC programs are still debatable.
Limitations
Sample Limitations
The sample for this study was selected from a single school in the Metro Atlanta Georgia
area. This was a Georgia suburban middle school in suburban Atlanta with 688 total students.
The student population was ethnically diverse (Black = 53%, White = 26%, Multi-Racial = 9%,
Asian = 7%, Other = 5%). The percentage of students eligible for free lunch was 32% and those
eligible for reduced lunch was 10%. The results of this study may not be generalizable to a)
schools outside of Metro Atlanta, Georgia, b) schools that are not ethnically diverse, or c)
schools that have a greater proportion of students eligible for the free lunch and reduced lunch
programs. Additionally, this study was only conducted among middle school students. It is not
known if the effects apply to lower or higher grades.
81
Statistical Power Limitations
The study was further limited by the sample size and the size of the effect. A power
analysis assuming a .8 effect size, .05 probability level, two groups, and one covariate yields a
sample size of 128. The effect size and probability levels are the desired standards for social
scientific research (Field, 2012; Pallant, 2013). The current sample of 189 was ample in size
based on the initial assumptions. However, the results indicated that the effect size was very
small, such that the power was only .136, which is much lower than the desired level of .80. As
a result, instead of there being an 80% chance of detecting a significant effect if one actually
existed in the real world, the chance was only 13.6%.
Methodological Limitation
A cut-off sampling approach is a possible threat to validity. Cut-off samples are not
representative of the overall population (Calonico, Cattaneo, & Titiunik, 2014; Cook, 2008). To
lessen the effect of the cut-off sampling approach impacting statistical tests that rely on
population estimates such as means or proportions, an ANCOVA was used. An ANCOVA
controls for the nonequivalent groups; the ANCOVA examines differences between two or more
groups on a continuous variable, while controlling for the effects of one or more variables (Ary
et al., 2010; Tabachnick & Fidell, 2012).
Ethical Procedures
This study was conducted based upon permission granted and the ethical standards
indicated by the Liberty University (See Appendixes A, School Approval and B, IRB Approval).
Following the standards of the Liberty University (IRB) ensured the ethical protection of all
research participants. Each participant’s confidentiality and anonymity was maintained. Data
was archived by the school. The researcher collated the relevant data. Data was archived
82
without nomenclature; student data was identifiable through computer generated identification
numbers. All student assessment data was stored according to the school’s policy. Research
data was stored securely online under the username and password of the researcher. Both during
the data analysis and after the final completion of the research, all was and will continue to be
conducted under secure processes. However, the data will be kept by the researcher indefinitely.
Recommendations for Future Research
Sample Recommendations
Given the study’s limitation due to the sample, I would recommend that the study be
replicated in the future with a larger sample size, as the effect was found to be very small. To
accurately calculate the needed sample size, the effect size measures should be set to small
instead of medium, maybe even very small, given the results of an eta square value of .004.
Additionally, future studies should also include samples from other school districts within and
outside of Georgia. The variation in sample also needs to include schools that are less diverse
ethnically, as well more economically challenged, exceeding the 32%/10% free/reduced lunch
ratio. High school students should also be examined as a sample population, along with younger
elementary school aged children.
Instrument Recommendation
This study was limited to comparing students on the AIMSweb test, which is a
standardized math test. Future research can be conducted in other subject matter areas with other
standardized measurements, as a limitation of this study was that it may be only generalizable to
those in the eighth grade math subject area; other skills could include English, reading, and
writing ability. Future studies should be conducted in English, reading, and writing to determine
83
if early identification of low performance in English, reading, and writing can also be corrected
within one school year.
84
REFERENCES
Adams, C. (2011). The effects of a remedial math intervention on standardized test scores in
Georgia middle schools (Doctoral dissertation). Liberty University, Lynchburg, VA.
Ai, X. (2002). Gender differences in growth in mathematics achievement: Three-level
longitudinal and multilevel analyses of individual, home, and school influences.
Mathematical Thinking & Learning, 4(1), 1-22. doi:10.1207/S15327833MTL0401_1
Aiken, L. S., West, S. G., Schwalm, D. E., Carroll, J., & Hsuing, S. (1998). Comparison of a
randomized and two experimental designs in a single outcome evaluation: Efficacy of a
university-level remedial writing program. Evaluation Review, 22, 207-244.
doi: 10.1177/0193841X9802200203
AIMSweb. (2009a). AIMSweb progress monitoring and improvement system.
Retrieved from http://www.aimsweb.com/
AIMSweb. (2009b). Mathematics concepts and applications: Administration and technical
manual. San Antonio, TX: Pearson, Inc.
Allensworth, E. M. (2005). Dropout rates after high-stakes testing in elementary school: A study
of the contradictory effects of Chicago's efforts to end social promotion. Educational
Evaluation and Policy Analysis, 27, 341-364. doi:org/10.3102/01623737027004341
Allensworth, E. M., & Nagaoka, J. (2010). Issues in studying the effects of retaining students
with high-stakes promotion tests: Findings from Chicago. In J. L. Meece & J. S. Eccles
(Eds.), Handbook of research on schooling, and human development (pp. 327-341). New
York, NY: Routledge.
Anagnostopoulos, D. (2006). "Real students" and "true demotes": Ending social promotion and
85
the moral ordering of urban high schools. American Educational Research Journal, 43,
5-42. doi:org/10.3102/00028312043001005
Angrist, J. D., & Lavy, V. (1999). Using Maimonides’ rule to estimate the effect of class size on
scholastic achievement. Quarterly Journal of Economics, 114(2), 533-575.
doi:10.1162/003355399556061
Archer, A. L., & Hughes, C. A. (2011). Explicit instruction: Effective and efficient teaching. In
K. R. Harris and S. Graham (Eds.), What works for special-needs learners. New York,
NY: Guilford Press.
Ary, D., Jacobs, L. C., Razavieh, A., & Sorensen, C. (2010). Introduction to research in
education (8th ed.). Belmont, CA: Thomson Wadsworth.
Atherton, J. (2013). Learning and teaching; Piaget’s developmental theory. Retrieved
from http://www.learningandteaching.info/learning/piaget.htm
Axtell, P. K., McCallum, R. S., Mee Bell, S. & Poncy, B. (2009). Developing math
automaticity using a classwide fluency building procedure for middle school students: A
preliminary study. Psychology in Schools, 46, 526–538. doi:10.1002/pits.20395
Baker, J. D., Rieg, S. A., & Clendaniel, T. (2006). An investigation of an after school
math tutoring program: University tutors + elementary students = a successful
partnership. Education, 127(2), 287-293.
Balfanz, R., & Byrnes, V. (2006). Closing the mathematics achievement gap in high-poverty
middle schools: Enablers and constraints. Journal of Education for Students Placed at
Risk, 11(2), 143–159.
Bandura, A. (1997). Self-efficacy: The exercise of control. New York: W.H. Freeman and
Company.
86
Barbash, S. (2012). Clear teaching: With direct instruction, Siegfried Engelmann discovered a
better way of teaching. Arlington, VA: Education Consumers Foundation.
Barton, P. (2004). Why does the gap persist? Educational Leadership, 62(3), 9-13.
Beatty, B. (2012a). Reliving the history of compensatory education: Policy choices,
bureaucracy, and the politicized role of science in the evolution of head start. Teachers
College Record, 114(6), 1-10. Retrieved from http://www.tcrecord.org ID Number:
16694
Beatty, B. (2012b). Rethinking compensatory education: Historical perspectives on race, class,
culture, language, and the discourse of the “disadvantaged child.” Teachers College
Record, 114(6), 1-11. Retrieved from http://www.tcrecord.org
Bennett, R. E., Rock, D. A., & Wang, M. (1991). Equivalence of free-response and multiple-
choice items. Journal of Educational Measurement, 28, 77–92.
Bennett, A., Bridgall, B. L., Cauce, A. M., Everson, H. T., Gordon, E. W., & Lee, C. D.
(2004). All students reaching the top: Strategies for closing academic achievement gaps.
Naperville, IL: Learning Point Associates, North Central Regional Educational
Laboratory.
Berk, R. A., & Rauma, D. (1983). Capitalizing on nonrandom assignment to treatments: A
regression discontinuity evaluation of a crime control program. Journal of the American
Statistical Association, 78(381), 21–27.
Berk, R., Barnes, G., Ahlman, L., & Kurt, E. (2010). When second best is good enough: A
comparison between a true experiment and a regression discontinuity experiment.
Journal of Criminology, 6(2), 191–208. doi:10.1007/s11292-010-9095-3
Bicknell, L. (2009). Curriculum implementation: A study of the effect of a specialized
87
curriculum on sixth grade mathematics summative test scores in a rural middle school
(Doctoral dissertation). Liberty University, Lynchburg, VA.
Bishop, A., & Forgsaz, H. (2007). Issues in access and equity in mathematics education. In F.
Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol.
2, pp.1145-1167). Charlotte, NC: Information Age Publishing.
Blankstein, A. (2013). Failure is not an option: 6 principles for making student success the
ONLY option (3rd ed.). Thousand Oaks, CA: Corwin.
Bloom, B. S. (Ed.). (1956). Taxonomy of educational objectives: The classification of
educational goals. Handbook 1, Cognitive domain. New York: McKay.
Boston, C. (2002). The concept of formative assessment. Practical Assessment, Research &
Evaluation, 8(9). Retrieved from http://PAREonline.net/getvn.asp?v=8&n=9
Bourdieu, P., & Passeron, J. C. (1990). Reproduction in education, society and culture (R. Nice,
Trans. 2nd ed.). London, England: Sage. (Original work published 1970)
Bottge, B. A., Heinrichs, M., Mehta, Z., Rueda, E., Hung, Y., & Danneker, J. (2004).
Teaching mathematical problem solving to middle school students in math
technology education, and special education classrooms. RMLE Online: Research
in Middle Level Education, 27(1), 1-17.
Bramlett, R., Cates, G., Savina, E., & Lauinger, B. (2010). Assessing effectiveness and
efficiency of academic interventions in school psychology journals: 1995–2005.
Psychology in the Schools, 47(2), 114-125. doi: 10.1002/pits.20457
Brenneman, R. (2015). Does differentiation work? A debate over instructional practices.
88
Education Week Teacher. Retrieved from
http://blogs.edweek.org/teachers/teaching_now/2015/02/does-differentiation-work-an-
instruction-debate.html
Brown, C. P. (2007). Examining the streams of a retention policy to understand the politics of
high stakes reform. Education Policy Analysis Archives, 15(9). Retrieved from
http://epaa.asu.edu/epaa/v15n9/
Brown, K. M. (2010). Schools of excellence and equity? Using equity audits as a tool to expose a
flawed system of recognition. International Journal of Education Policy and Leadership,
5(5), 15-22. Retrieved from www.ijepl.org.
Brown, M., Benkovitz, J., Muttillo, A., & Urban, T. (2011). Leading schools of excellence and
equity: Documenting effective strategies in closing achievement gaps. Teachers College
Record, 113(1), 57-96. Retrieved from
https://nces.ed.gov/nationsreportcard/pubs/studies/2013451.aspx
Brown v. Board of Education (1954). Retrieved from
http://www.civilrights.org/education/brown/brown.html
Burns, M. K., & Gibbons, K. (2012). Response to intervention implementation in elementary
and secondary schools: Procedures to assure scientific-based practices (2nd ed.). New
York: Routledge.
Caldwell, J. (2008). Comprehension assessment: A classroom guide. New York: Guilford
Press.
Calonico, S., Cattaneo, M., & Titiunik, R. (2014). Robust data-driven inference in the
regression-discontinuity design. The Stata Journal, 14(4), 909–946.
doi:10.3982/ECTA11757
89
Carroll, J. B. (1989). The Carroll Model: A 25-year retrospective and prospective view.
Educational Researcher, 18(1), 26-3. Retrieved in ERIC (EJ386602)
Carr, M., & Alexeev, N. (2011). Fluency, accuracy, and gender predict developmental
trajectories of arithmetic strategies. Journal of Educational Psychology, 103, 617–631.
doi:10.1037/a0023864
Caygill, R., & Eley, L. (2001). Evidence about the effects of assessment task format on
student achievement. Paper presented at the Annual Conference of the British
Educational Research Association, University of Leeds, England. Retrieved from
http://www.leeds.ac.uk/educol/documents/00001841.htm
Chan, J. C. K. (2010). Long-term effects of testing on the recall of nontested materials.
Memory, 18(1), 49-57. doi:10.1080/09658210903405737
Chapman, K., Tatiana, J., Hartlep, N., Vang, M., & Lipsey, T. (2014). The double-edged sword
of curriculum: How curriculum in majority White suburban high schools supports and
hinders the growth of students of color. Curriculum & Teaching Dialogue, 16(1-2), 1-87.
Claessens, A., Engel, M., & Curran, F. (2013). Academic content, student learning, and
persistence of preschool effects. American Educational Research Journal, 50(6).
doi:10.3102/0002831213513634
Clements, H., & Sarama, J. (2011). Early childhood mathematics intervention. Science,
333(6045), 1-14. doi:10.1126/science.1204537
Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Hillsdale, NJ:
Lawrence Erlbaum.
Committee on Science and Technology, U.S. House of Representatives. (2007). President signs
Gordon’s COMPETES Act. Retrieved from
90
http://science.house.gov/press/PRArticle.aspx?NewsID=1945
Cook, Thomas D. (2008). "‘Waiting for Life to Arrive’: A history of the regression-discontinuity
design in Psychology, Statistics and Economics." Journal of Econometrics 142(2), 636–
654.
Cook, T. D., & Campbell, D. T. (1979). Quasi-experimentation: Design & analysis issues for
field settings. Boston, MA: Houghton Mifflin.
Cortes, K., Goodman, J., & Nomi, T. (2013). A double dose of algebra. Education Next, 13(1),
70-76. Retrieved from http://educationnext.org/files/ednext_20131_cortes.pdf.
Cortes, K., Goodman, J., & Nomi, T. (2015). Intensive math instruction and educational
attainment: Long run impacts of double-dose algebra. Journal of Human Resources,
50(1), 108–158. Retrieved from https://aefpweb.org/sites/default/files/webform/Cortes-
Goodman-Nomi-RemedialMath%2003-04-2012.pdf
Cosier, M., Causton-Theoharis, J., & Theoharis, G. (2013). Does access matter? Time in general
education and achievement for students with disabilities. Remedial and Special
Education, 34(6), 323-332. doi:10.1177/0741932513485448
Courtade, G., Spooner, F., Browder, D., & Jimenez, B. (2012). Seven reasons to promote
standards based instruction for students with severe disabilities: A reply to Ayres,
Lowrey, Douglas, & Sievers (2011). Education and Training in Autism and
Developmental Disabilities, 47(1), 3–13. Retrieved from
http://daddcec.org/Portals/0/CEC/Autism_Disabilities/Research/Publications/Education_
Training_Development_Disabilities/2011v47_journals/ETADD_2012v47n1p3-
13_Seven_reasons.pdf
Crane, J. (2002). The promise of value-added testing. Progressive Policy Institute.
91
Retrieved from www.ppionline.org
Cusumano, D. L. (2007). Is it working? An overview of curriculum based measurement and its
uses for assessing instructional, intervention, or program effectiveness. The Behavior
Analyst Today, 8, 24-34. Retrieved in ERIC (EJ800967)
Dawn Leslie, D., & Mendick, H. (2013). Debates in mathematics education. New York:
Routledge.
Dembosky, J., Pane, J., Barney, H., & Christina, R. (2006). Data driven decision-making
in southwestern Pennsylvania school districts. Retrieved from
http://www.rand.org/pubs/working_papers/2006/RAND_WR326.pdf.
Din, C. S., Song, K., & Richardson, L. (2006). Do mathematical gender differences
continue? A longitudinal study of gender difference and excellence in
mathematics performance. U.S. Educational Studies, 40(3), 279-295.
Donald, K. (2009). Evaluation of self-reported teacher efficacy and minority achievement in
middle school (Doctoral dissertation). Capella University, Minneapolis, MN.
Domina, T. (2014). The link between middle school mathematics course placement and
achievement. Child Development, 85, 1948–1964. doi:10.1111/cdev.12255
Dougherty, S., Goodman, J., Litke, E., & Page, L. (2015). Middle school math acceleration and
equitable access to eighth-grade algebra: Evidence from the Wake County Public School
System. Educational Evaluation and Policy Analysis 37(1), 80–101.
doi:10.3102/0162373715576076
Duffrin, E., & Scott, C. (2008). Uncharted territory: An examination of restructuring under
NCLB in Georgia. Washington, DC: Center on Education Policy.
Durwood, C., Krone, E., & Mazzeo, C. (2010). Are two algebra classes better than one? The
92
effects of double-dose instruction in Chicago. Chicago, IL: Consortium on Chicago
School Research. Retrieved from
http://ccsr.uchicago.edu/sites/default/files/publications/Double%20 Dose-
7%20Final%20082610.pdf.
Eckert, T., Dunn, E., Codding, R., Begeny, J., & Kleinmann, A. (2006). Assessment of
mathematics and reading performance: An examination of the correspondence between
direct assessment of student performance and teacher report. Psychology in the Schools,
43(3), 247-265. doi:10.1002/pits.20147
Edmonds, W., & Kennedy, T. (2013). An applied reference guide to research designs:
quantitative, qualitative, and mixed methods. London: SAGE.
Educational Commission of the States. (2005). Student promotion/retention policies. Retrieved
from http://www.ecs.org/clearinghouse/65/51/6551.pdf
Engel, M., Claessens, A., & Finch, M. (2013). Teaching students what they already know?
The (mis)alignment between instructional content in mathematics and student knowledge
in kindergarten. Educational Evaluation and Policy Analysis, 35(2), 157-178.
doi:10.3102/0162373712461850
Ertmer, P. A., & Newby, T. J. (2013). Behaviorism, cognitivism, constructivism: Comparing
critical features from an instructional design perspective. Performance Improvement
Quarterly, 26(2), 43-71. doi: 10.1002/piq.21143
Eskin, L. (2004). Triumph learning. Retrieved from
http://www.triumphlearning.com/Pages/articles/TL_ValidationStudy_Eskin.pdf
Espin, C. A., Scierka, B. J., Skare, S., & Halverson, N. (1999). Criterion-related validity of
93
curriculum-based measures in writing for secondary school students. Reading and
Writing Quarterly: Overcoming Learning Difficulties, 15, 5-27.
Farbman, D., Christie, K., Davis, J., Griffith, M., & Zinth, J. D. (2011). Learning time in
America: Trends to reform the American school calendar. Boston, MA: National Center
on Time & Learning.
Field, A. (2012). Discovering Statistics using IBM SPSS Statistics. London: SAGE.
Finn, J., Gerber, S., & Boyd-Zaharias, J. (2005). Small classes in the early grades, academic
achievement, and graduating from high school. Journal of Educational Psychology,
97(2), 214-223. doi:10.1037/0022-0663.97.2.214
Finnigan, K. S., & Gross, B. (2007). Do accountability policy sanctions influence teacher
motivation? Lessons from Chicago's low-performing schools. American Educational
Research Journal, 44, 594-629. doi:org/10.3102/0002831207306767
Fisher, D., & Frey, N. (2007). Checking for understanding formative assessment
techniques for your classroom. Alexandria, VA: Association for Supervision and
Curriculum Development.
Flores, M. M., & Kaylor, M. (2007). The effects of a direct instruction program on the fraction
performance of middle school students at-risk for failure in mathematics. Journal of
Instructional Psychology, 34(2), 84-94. Retrieved in ERIC (EJ774167)
Foegen, A., Jiban, C., & Deno, S. L. (2007). Progress monitoring measures in
mathematics: A review of the literature. The Journal of Special Education, 41(2),
121-139. doi: 10.1177/00224669070410020101
Forgasz, H., & Rivera, F. (2012). Towards equity in mathematics education. Berlin: Springer-
Verlag.
94
Fram, S. M., Miller-Cribbs, J. E., Horn, V., & Lee, L. (2007). Poverty, race, and the
contexts of achievement: Examining educational experiences of children in the U.S.
south. Social Worker, 52(4), 309-319. Retrieved from
http://www.citeulike.org/article/1795830
Fredriksson, P., Ockert, B., & Oosterbeek, H. (2013). Long-term effects of class size. Quarterly
Journal of Economics, 128(1), 249-285. doi: 10.1093/qje/qjs048
Fullan, M., Hill, P., & Crevola, C. (2006). Breakthrough. Thousand Oaks, CA: Corwin Press.
Gamble, B. E., Kim, S., & An, S. (2012). Impact of a middle school math academy on learning
and attitudes of minority male students in an urban district. Long Beach, CA: California
State University, Long Beach.
Ganley, C., Mingle, L., Ryan, A., Ryan, K., Vasilyeva, M., & Perry, M. (2013). An examination
of stereotype threat effects on girls’ mathematics performance. Developmental
Psychology, 49(10), 1886-1897. Retrieved from http://dx.doi.org/10.1037/a0031412
Gall, M., Gall, J., & Borg, W. (2007). Educational research: An introduction. Boston:
Pearson/Allyn & Bacon.
Gardner, D. (1983). A nation at risk: The imperative for educational reform—Report of the
National Commission on Excellence in Education. Washington, DC: U.S. Government
Printing Office. Retrieved from
http://datacenter.spps.org/uploads/sotw_a_nation_at_risk_1983.pdf
Gardner, H. (1999). Intelligence reframed. Multiple intelligences for the 21st century.
New York: Basic Books.
Garner, M., & Engelhard, G. (1999). Gender differences in performance on multiple-choice and
95
constructed response mathematics items. Applied Measurement in Education, 12, 29-51.
doi:10.1207/s15324818ame1201_3
Georgia Department of Education. (2008). Response to intervention: Georgia’s student
achievement pyramid of interventions. Retrieved from
http://www.doe.k12.ga.us/ci_services.aspx?PageReq=CIServRTI.
Georgia Department of Education. (2009). State of Georgia consolidated state
application accountability workbook. Retrieved from
http://www.ed.gov/admins/lead/account/stateplans03/gacsa.pdf.
Georgia Department of Education. (2010). Education support and improvement.
Retrieved from http://www.doe.k12.ga.us/tss.aspx#.
Georgia Department of Education. (2011). Georgia Performance Standards (GPS).
Retrieved from
https://www.georgiastandards.org/Standards/Pages/BrowseStandards/BrowseGPS
.aspx.
Georgia State Board of Education. (2001). Official Code 20-2-282 through 20-2-285. Retrieved
from http://www.doe.k12.ga.us/External-Affairsand-Policy/Policy/Pages/Promotion-and-
Retention.aspx
Georgia GPS edition COACH standards-based instruction, math grade 8. (2008). New York:
Triumph Learning.
Geary, D., Hoard, M., Nugent, L., & Byrd-Craven, J. (2009). Strategy use, long-term
memory, and working memory capacity. In D. B. Berch & M. M. M. Mazzocco
(Eds.), Why is math so hard for some children? The nature and origins of
mathematical learning difficulties and disabilities (pp. 83-105). Baltimore, MD:
96
Brooks.
Gersten, R., Chard, D. J., Madhavi, J., Baker, S. K., Morphy, P., & Flojo, J. (2009).
Mathematics instruction for students with learning disabilities: A meta-analysis of
instructional components. Review of Educational Research, 79(3), 1202-1242.
Goddard, R. D., Hoy, W. K., & Woolfolk-Hoy, A. W. (2004). Collective efficacy beliefs:
Theoretical developments, empirical evidence, and future directions. Educational
Researcher, 33(3), 3-13. Retrieved from
http://www.greatschoolsnow.com/02ERv33n3-Goddard_Collective_Efficacy.pdf
Gonzalez, L., Frankson, D., & Shealey, M. (2008). Urban education research: A paradigm shift.
In M. S. Plakhotnik & S. M. Nielsen (Eds.), Proceedings of the Seventh Annual College
of Education Research Conference: Urban and international education section (pp. 46-
51). Miami, FL: International University. Retrieved from
http://coeweb.fiu.edu/research_conference/
Goodman, J. (2012). The labor of division: Returns to compulsory math coursework (Harvard
Kennedy School Faculty Research Working Paper Series, RWP12–RW032). Retrieved
from http://scholar. harvard.edu/files/joshuagoodman/files/rwp12- 032_goodman.pdf
Gordon, B. (2007). U.S. competitiveness: The education imperative. Issues in Science and
Technology, 23(3), 31-36. Retrieved from
http://search.proquest.com/openview/a67810c90dd67ee9d9082416642d9e61/1?pq-
origsite=gscholar
Graney, S., Missall, K., Martinez, R., & Bergstrom, M. (2009). A preliminary investigation of
within-year growth patterns in reading and mathematics curriculum-based measures.
Journal of School Pyschology, 47(2), 121-142. doi:10.1016/j.jsp.2008.12.001
97
Greene, J. P., & Winters, M. A. (2007). Revisiting grade retention: An evaluation of Florida's
test based promotion policy. Education Finance and Policy, 2, 319-340.
doi:org/10.1162/edfp.2007.2.4.319
Greene, J. P., & Winters, M. A. (2009). The effects of exemptions to Florida's test-based
promotion policy: Who is retained? Who benefits academically? Economics of Education
Review, 28, 135- 142. doi:org/10.1016/j.econedurev.2008.02.002
Griffin, P., & Nix, P. (1991). Educational assessment and reporting: A new approach. Issues in
Science and Technology, 23(3), 31-37.
Gundlach, M. (2012, February, 28). The roots of differentiated instruction in teaching.
(Web log post via Bright Hub Education). Retrieved from
http://www.brighthubeducation.com/teaching-methods-tips/106939-history-of-
differentiated-instruction/
Guskey, T. R., & Passaro, P. (1998). Teacher efficacy: A study of construct dimension
measurement and change. American Educational Research Association, 31, 627-643.
Retrieved in ERIC (ED422396)
Haladyna, T. M. (1999). Developing and validating multiple choice test items (2nd ed.).
Mahwah, NJ: Lawrence Erlbaum.
Hall, T., Strangman, N., & Meyer, A. (2011). Differentiated instruction and implications
for UDL implementation. Wakefield, MA: National Center on Accessible
Instructional Materials. Retrieved from
http://aim.cast.org/learn/historyarchive/backgroundpapers/differentiated_
instruction_udl
Harlacher, J., Nelson, N., & Sanford, A. (2010). The "I" in RTI: Research-based factors for
98
intensifying instruction. Teaching Exceptional Children, 42, 30-38. Retrieved from
http://pdxscholar.library.pdx.edu/cgi/viewcontent.cgi?article=1003&context=edu_fac
Hattie, J. A. (2009). Visible learning: A synthesis of over 800 meta-analyses relating
to achievement. New York: Routledge.
Hattie, J. A. (2011). Visible learning for teachers: Maximizing impact on learning. New
York: Routledge.
Henry, G. T., Rickman, D. K., Fortner, C. K., & Henrick, C. C. (2005). Report of the findings
from Georgia's third grade retention policy. Atlanta, GA: Andrew Young School of
Public Policy.
Hosp, M. K., Hosp, J. L., & Howell, K.W. (2007). The ABCs of CBM: A practical
guide to curriculum-based measurement. New York: Guilford Press.
Huddleston, A. (2014). Achievement at whose expense? A literature review of test-based grade
retention polices in U.S. schools. Educational Policy Analysis Archives, 22(18).
doi: 10.14507/epaa.v22n18.2014
Imbens, G., & Lemieux, T. (2008). Regression discontinuity designs: A guide to practice.
Journal of Econometrics, 142(2), 615-635..
Iorio, J., & Adler, S. (2013). Take a number, standard line, better yet, be a number get tracked:
The assault of longitudinal data systems on teaching and learning. Teachers College
Record. Retrieved from http://www.tcrecord.org ID Nu
Jacob, B., & Lefgren, L. (2004). Remedial education and student achievement: A regression-
discontinuity analysis. Review of Economics and Statistics, 68, 226–244. Retrieved from
http://files.eric.ed.gov/fulltext/ED465007.pdfc
Jacob, B., & Lefgren, L. (2009). The effect of grade retention on high school completion.
99
American Economic Journal: Applied Economics, 12, 1-45.
doi:10.1.1.184.2180&rep=rep1&type=pdf
Jiban, C., & Deno, S. (2007). Using math and reading curriculum-based measurements to
predict state mathematics test performance: Are simple one-minute measures
technically appropriate? Assessment for Effective Intervention, 32(2), 78-108.
doi:10.1177/1534508407032002050
Jenkins, J. R., Schiller, E., Blackorby, J., Thayer, S. K., & Tilly, W. D. (2013).
Responsiveness to intervention: Architecture and practices. Learning Disability
Quarterly, 36, 36-46. doi:10.1177/0731948712464963
Jitendra, A. K. (2013). Understanding and accessing standards-based mathematics for students
with mathematics difficulties. Hammill Institute on Disabilities, 36 (1), 4-8.
doi: 10.1177/0731948712455337
Johnson, F. U., Hedner, M., & Olsson, M. J. (2012). The testing effect as a function of explicit
testing instructions and judgments of learning. Experimental Psychology, 59, 251–257.
Judge, S., & Watson, S. M. R. (2011). Longitudinal outcomes for mathematics achievement for
students with learning disabilities. The Journal of Educational Research, 104, 147–157.
doi:10.1080/00220671003636729
Kagan, S. (1994). Cooperative learning. San Clemente, CA: Kagan Press.
Kane, J., & Mertz, J. (2012). Debunking myths about gender and mathematics performance.
Notices of American Mathematics Society, 59(1), 10-21. doi:org/10.1090/noti790
Katz, I. R., Bennett, R. E., & Berger, A. E. (2000). Effects of response format on difficulty
of SAT-Mathematics items: It’s not the strategy. Journal of Educational Measurement,
37, 39–57.
100
Klein, D. (2003). A brief history of American K-12 mathematics education in the 20th
century. Mathematical Cognition. Retrieved from
http://www.csun.edu/~vcmth00m/AHistory.html
Krawec, J. (2013). Problem representation a mathematical problem-solving of students of
varying math ability. Learning Disability Quarterly, 36, 80-92.
doi:10.1177/0022219412436976
Krawec, J., Huang, J., Montague, M., Benikia, B., & Melia de Alba, A. (2012). The effects of
cognitive strategy instruction on knowledge of math problem-solving processes of middle
school students with learning disabilities. Learning Disability Quarterly, 36(2), 80-92.
doi:10.1177/0731948712463368
Ku, K. Y. L. (2009). Assessing students’ critical thinking performance: Urging for measurements
using multi-response format. Thinking Skills and Creativity, 4, 70–76.
La Morte, M. (2005). School law cases and concepts. Boston: Pearson.
Ladd, H., & Walsh, R. (2002). Implementing value-added measures of school effectiveness:
Getting the incentives right. Economics of Education Review, 21(1), 1-17.
doi:10.1016/S0272-7757(00)00039-X
Lareau, A. (2003). Unequal childhoods: Class, race, and family life. Berkeley, CA: University of
California Press.
Lauer, P., Akiba, M., Wilkerson, S., Apthorp, H., Snow, D., & Martin-Glenn, M. (2006).
Out-of-school time programs: A meta-analysis of effects for at-risk students. Review of
Educational Research, 76(2), 275-313. Retrieved from
http://www.sagepub.com/vaughnstudy/articles/children-youth/Lauer.pdf
Lembke, E. S., Hampton, D., & Beyers, S. J. (2012). Response to intervention in mathematics:
101
Critical elements. Psychology in the Schools, 49, 257–272. doi:10.1002/pits.21596K
Lindberg, S. M., Hyde, J. S., Petersen, J. L., & Linn, M. C. (2010). New trends in gender and
mathematics performance: A meta-analysis. Psychological Bulletin, 136(6), 1123-1135.
Retrieved from http://dx.doi.org/10.1037/a0021276
Linn, R. L., Baker, E. L., & Dunbar, S. B. (1991). Complex, performance-based assessment:
expectations and validation criteria. Educational Researcher, 20(8), 15-21.
doi:10.3102/0013189X020008015
Linn, R., Baker, E., & Betebenner, D. (2002). Accountability systems: Implications of
requirements of the No Child Left Behind Act of 2001. Educational Researcher, 31(6), 3-
16. doi:10.3102/0013189X031006003
Lissitz, R. W. (2014). Value added modeling and growth modeling with particular application to
teacher and school effectiveness. Charlotte: Information Age Publishing.
Livingston, D. R., & Livingston, S. M. (2002). Failing Georgia: The case against the ban on
social promotion. Education Policy Analysis Archives, 10(49). Retrieved from
http://epaa.asu.edu/ojs/article/view/328
Lukas, S., & Beresford, L. (2010). Naming and classifying: Theory, evidence, and equity in
education. In A. Luke, J. Green, & G. Kelly (Eds.), Review of research in education (Vol.
35, pp. 25–84). Washington: American Educational Research Association.
Lukhele, R., Thissen, D., & Wainer, H. (1994). On the relative value of multiple-choice,
constructed response, and examinee selected items on two achievement tests. Journal of
Educational Measurement, 31, 234–250. Retrieved from
http://www.jstor.org/stable/1435268
Maker, C. J. & Schiever, S. W. (2005). Teaching models in education of the gifted. Austin, TX:
102
Pro-ed.
Malmgren, K., McLaughlin, M., & Nolet, V. (2005). Accounting for the performance of
students with disabilities on statewide assessments. Journal of Special Education, 39(2),
86-96.
Martinez, S. (2012). From “culturally deprived” to “at risk”: The politics of popular expression
and educational inequality in the United States, 1960-1985. Teachers College Record,
114(6), 1-31. Retrieved from http://www.tcrecord.org ID Number: 16691
Martinez, M., & McGrath, D. (2014). Deeper learning: How eight innovative public schools
are transforming education in the twenty-first century. New York: New Press. Retrieved
from http://www.ebrary.com
Marsh, J., Pane, J., & Hamilton, L. (2006). Making sense of data-driven decision making in
education. Evidence from recent RAND research. Retrieved from
http://www.rand.org/pubs/occasional_papers/2006/RAND_OP170.pdf
Marzano, R. (2003). What works in schools: Translating research into action.
Alexandria, VA: Association for Supervision and Curriculum Development.
Marzano, R. J., Pickering, D. J., & Pollock, J. E. (2001). Classroom instruction that works:
Researched based strategies for increasing student achievement. Alexandria, VA:
Association for Supervision and Curriculum Development.
Matsudaira, J. (2007). Mandatory summer school and student achievement. Journal of
Econometrics, 142, 829-850. doi:10.1016/j.jeconom.2007.05.015
McCombs, J. S., Kirby, S. N., & Mariano, L. T. (Eds.). (2009). Ending social promotion without
leaving children behind: The case of New York City. Santa Monica, CA: RAND.
McDowell, K., Lonigan, C., & Goldstein, H. (2007). Relations among socioeconomic status,
103
age, and predictors of phonological awareness. Journal of Speech, Language, and
Hearing Research, 50, 1079-1092. doi:10.1044/1092-4388(2007/075)
Methe, S., Kilgus, S., Neiman, C., & Riley-Tillman, C. (2012). Meta-Analysis of interventions
for basic mathematics computation in single-case research. Journal of Behavioral
Education, 21(3), 230-253. doi:10.1007/s10864-012-9161-1
Mojavezi, A., & Tamiz, M. (2012). The impact of teacher self-efficacy on the students’
motivation and achievement. Theory and Practice in Language Studies, 2(3), 483-491.
doi:10.4304/tpls.2.3.483-491
Mordica, J. (2006). Third grade students not meeting standards in reading: A longitudinal
study. (Research Brief No. 02 ). Atlanta, GA: Georgia Department of Education
Munk, J., Gibb, G., & Caldarella, P. (2010). Collaborative Preteaching of Students at Risk for
Academic Failure. Intervention in School and Clinic, 45(3), 177-185 doi:
10.1177/1053451209349534
National Academy of Sciences, National Academy of Engineering, & Institute of
Medicine. (2005). Rising above the gathering storm: Energizing and employing
America for a brighter economic future. Retrieved from
http://sciencedems.house.gov/Media/File/Reports/natacad_compete_exsum_6feb06.pdf
National Center for Education Statistics. (2009). Achievement gap. Retrieved from
http://search.nces.ed.gov/search?output=xml_no_dtd&site= nces&client
=nces&proxystylesheet=nces&q=achievement+gap
National Education Association. (2006). Tomorrow's teachers: Help wanted: Minority
teachers. Retrieved from
http://www.nea.org/tomorrowsteachers/2002/helpwanted.html.
104
National Mathematics Advisory Panel. (2008). The final report of the National Mathematics
Advisory Panel. Retrieved from
http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf
Neal, D., & Schanzenbach, D. W. (2010). Left behind by design: Proficiency counts and test-
based accountability. Review of Economics and Statistics, 92, 263-283.
doi:org/10.1162/rest.2010.12318
Noell, G. (2005). Assessing teacher preparation program effectiveness: A pilot examination.
Retrieved from www.asa.regents.state.la.us/TE/digest.pdf
Nomi, T. (2012). The unintended consequences of an algebra-for-all policy on high-skill
students: Effects on instructional organization and students’ academic outcomes.
Educational Evaluation and Policy Analysis, 34, 489–505.
doi:10.3102/0162373712453869
Nomi, T., & Allensworth, E. (2009). Double dose algebra as an alternative strategy to
remediation: Effects on students’ academic outcomes. Journal of Research on
Educational Effectiveness, 2(2), 111–148. doi:10.1080/19345740802676739
Nomi, T., & Allensworth, E. M. (2011) Double-dose algebra as a strategy for improving
mathematics achievement of struggling students: Evidence from Chicago Public Schools.
In R. R. Gersten & B. B. Newman-Gonchar (Eds.), Response to intervention in
mathematics. Baltimore, MD: Brookes Publishing.
Nomi, T., & Allensworth, E. (2013). Sorting and supporting: Why double-dose algebra leads to
better test scores but more course failures. American Educational Research Journal,
50(4), 756–788. Retrieved from http://www.jstor.org/stable/232526104
Noonan, B., & Duncan, C. (2005). Peer and self-assessment in high school. Practical
105
Assessment, Research & Evaluation, 10(17). Retrieved from
http://pareonline.net/getvn.asp?v=10&n=17
Oketch, M., Mutisya, M., Sagwe, J., Musyoka, P., & Ngware, M. W. (2012). The effect of active
teaching and subject content coverage on students’ achievement: Evidence from primary
schools in Kenya. London Review of Education, 10(1), 19-33.
doi:10.1080/14748460.2012.659057
Olejnik, S., & Algina, J. (2000). Measures of effect size for comparative studies: applications,
interpretations, and limitations. Contemporary Educational Psychology, 25, 241–286.
Orfield, G., Losen, D., Wald, J., & Swanson, C. (2004). Losing our future: How minority youth
are being left behind by the graduation rate crisis. Cambridge, MA: The Civil Rights
Project at Harvard University. Retrieved from http://www.civilrightsproject.harvard.edu
Owen, S. A., & Ranick, D. L. (1977). The Greensville program: A commonsense approach to
basics. Phi Delta Kappan, 58, 531-533, 539.
Pallant J. (2013). SPSS survival manual, a step by step guide to data analysis using IBM
SPSS (5th ed). New York: McGraw Hill.
Patall, E. A., Cooper, H., & Allen, A. B. (2010). Extending the school day or school year: A
systematic review of research (1985–2009). Review of Educational Research, 80(3), 401–
436.
Perie, M., Grigg, W. S., & Donahue, P. L. (2005). The Nation’s report card: Reading
2005 (NCES 2006–451). U.S. Department of Education, Institute of Education
Sciences, National Center for Education Statistics. Washington, DC: U.S.
Government Printing Office.
Petscher, Y., Young-Suk, K., & Foorman, B. R. (2012). The importance of predictive
106
power in early screening assessments: Implications for placement in the response
to intervention framework. Assessment for Effective Intervention, 36(3), 158-166.
doi: 10.1177/1534508410396698
Phelan, J., Choi, K., Vendlinski, T., Baker, E., & Herman, J. (2011). Differential improvement
in student understanding of mathematical principles following formative assessment
intervention. Journal of Educational Research, 104(5), 330-339.
doi:10.1080/00220671.2010.484030
Porter, A., McMaken, J., Hwang, J., & Yang, R. (2011). Common Core standards: The new U.S.
intended curriculum. Educational Research, 40, 103–116.
doi:10.3102/0013189×11405038
Posner, G., & Rudnitsky, A. (2006). Course design a guide to curriculum development for
teachers. New York: Pearson.
Powell, S. (2012). High-stakes testing for students with mathematics difficulty: Response
format effects in mathematics problem solving. Learning Disability Quarterly, 35(1), 3-
9. doi:10.1177/0731948711428773
Powell, S., Fuchs, L., & Fuchs, D. (2013). Reaching the mountaintop: addressing the common
core standards in mathematics for students with mathematics difficulties. Learning
Disabilities Research & Practice, 28(1), 38–48. doi: 10.1111/ldrp.12001
Pupel, D. (2001). Moral outrage in education. New York: Peter Lang.
Ready, R., Edley Jr., T., & Snow, C. E. (Eds.). (2002). Achieving high educational
standards for all: Conference summary. Washington, DC: National Academy Press.
Redd, Z., Boccanfuso, C., Walker, K., Princiotta, D., Knewstub, D., & Moore, K. (2012).
107
Expanding time for learning both inside and outside of the classroom: A review of the
evidence base. Washington, DC: Child Trends. Retrieved from
http://www.childtrends.org/wp-content/ uploads/2013/03/Child_Trends-
2012_08_16_RB_TimeForLearning.pdf.
Riccomini, P. J., & Witzel, B. S. (2009). Response to intervention in math. Chicago: Corwin
Press.
Rickles, J. H. (2013). Examining heterogeneity in the effect of taking algebra in eighth grade.
The Journal of Educational Research, 106, 251–268.
doi:10.1080/00220671.2012.692731
Rivkin, S. G. (2007). Value-added analysis and education policy. Calder Urban Institute (1).
National Center for Analysis of Longitudinal Data in Education Research. Tracking every
student’s learning every year. Retrieved from
http://www.urban.org/UploadedPDF/411577_value-added_analysis.pdf
Roediger, H. L., Agarwal, P. K., Kang, S. H. K., & Marsh, E. J. (2010). Benefits of testing
memory: Best practices and boundary conditions. In G. M. Davies & D. B. Wright
(Eds.), New frontiers in applied memory (pp. 13–49). Brighton: Psychology Press.
Roderick, M., Bryk, A., & Jacob, B. (2002). The impact of high-stakes testing in Chicago on
student achievement in promotional gate grades. Educational Evaluation & Policy
Analysis, 24(4), 333-57.
Roderick, M., & Nagaoka, J. (2005). Retention under Chicago's high-stakes testing program:
Helpful, harmful, or harmless? Educational Evaluation and Policy Analysis, 27, 309-340.
doi:org/10.3102/01623737027004309
Rojas-LeBouef, A., & Slate, J. (2012). The achievement gap between white and non-white
108
students. International Journal of Educational Leadership Preparation, 7(1), 1-66.
Ross, J. A. (1994). Beliefs that make a difference: The origins and impacts of
teacher efficacy. Paper presented at the annual meeting of the Canadian
Association for Curriculum Studies. Retrieved in ERIC (ED379216)
Ross, S. M., Sanders, W., Wright, S., Stringfield, S., Wang, L., Weiping, A., & Albert, M.
(2001). Two and three year achievement results from the Memphis restructuring
initiative. School Effectiveness and School Improvement, 12(3), 323-46. Retrieved in
ERIC (EJ634730).
Rosenzweig, C., Krawec J., & Montaque, M. (2011). Metacognitive strategy use of eighth-grade
students with and without learning disabilities during mathematical problem solving: a
think-aloud analysis. Journal Learning Disabilities, 44(6), 508-520. doi:
10.1177/0022219410378445
Rossi, P., Lipsey, M., & Freeman, H. (2004). Evaluation: A systematic approach (7th ed).
Thousand Oaks, CA: Sage Publications.
Rothstein, R. (2004). The achievement gap: A broader picture. Educational Leadership, 62(3),
40-43. Retrieved from
http://www.ascd.org/publications/educational-leadership/nov04/vol62/num03/The-
Russell, S. J. (2012). CCSSM: Keeping teaching and learning strong. Teaching Children
Mathematics, 19, 50–56. Retrieved from
http://www.jstor.org/stable/10.5951/teacchilmath.19.1.0050
Safer, N., & Fleischman, S. (2005). Research matters: How student progress monitoring
109
improves instruction. How Schools Improve, 62(5), 81-83. Retrieved from
http://studentprogress.org/library/ArticlesResearch/Edleadershiparticle.pdf
Schatschneider, C., Petscher, Y., & Williams, K. (2008). How to evaluate a screening
process: The vocabulary of screening and what educators need to know. In L. Justice &
C. Vukelich (Eds.), Achieving excellence in preschool literacy instruction (pp. 304-316).
New York: Guilford Press.
Schellings, G. L. M., & Broekkamp, H. (2011). Signaling task awareness in think-aloud
protocols from students selecting relevant information from text. Metacognition
Learning, 6, 65-82. doi: 10.1007/s11409-010- 9067-z
Schmoker, M. (2006). Results now: How we can achieve unprecedented improvements in
teaching and learning. Alexandria, VA: Association for Supervision and Curriculum
Development.
Schrank, F. A., & Wendling, B. J. (2009). Educational interventions and accommodations
related to the Woodcock-Johnson® III Tests of Cognitive Abilities and the Woodcock-
Johnson III Diagnostic Supplement to the Tests of Cognitive Abilities (Woodcock-
Johnson III Assessment Service Bulletin No. 10). Rolling Meadows, IL: Riverside
Publishing. Retrieved from
http://www.riversidepublishing.com/clinical/pdf/WJIII_ASB10.pdf
Sezer, R. (2010). Pulling out all stops. Education, 130(3), 416-423.
Shadish, W. R., Cook, T. D., & Campbell, D. T. (2002). Experimental and quasi-experimental
designs for generalized casual inference. Boston, MA: Houghton Mifflin.
Shadish, W. R., Galindo, R., Wong, V. C., Steiner, P. M., & Cook, T. D. (2011). A randomized
experiment comparing random and cutoff-based assignment. Psychological Methods,
110
16(2), 179-191. doi:10.1037/a0023345
Shapiro, E. (2011). Academic skills problems: Direct assessment and intervention. New York:
Guilford Publications.
Shapiro, E. S., & Gebhardt, S. N. (2012). Comparing computer-adaptive and curriculum-based
measurement methods of assessment. School Psychology Review, 41(3), 295-305.
Shapiro, E. S., Keller, M. A., Lutz, J. G., Santoro, L. E., & Hintz, J. M. (2006).
Curriculum-based measures and performance on state assessment and standardized tests.
Journal of Psychoeducational Assessment, 24(1), 19-35.
Shinn, M. R. (2008). Best practices in using curriculum-based measurement in a problem-solving
model. In Thomas & Grimes (Eds.), Best practices in school psychology V. Bethesda,
MD: National Association of School Psychologists.
Siegler, R. S., Duncan, G. J., Davis-Kean, P. E., Duckworth, K., Claessens, A., Engel, M., . . .
Chen, M. (2012). Early predictors of high school mathematics achievement.
Psychological Sciences, 23, 691–697. doi:10.1177/0956797612440101
Silva, E. (2007). On the clock: Rethinking the way schools use time. Washington, DC: Education
Sector.
Silverman, F. (2004). Helping minority students after Brown v. Board of Education.
District Administration, 40(7), 17. Retrieved from
http://connection.ebscohost.com/c/articles/13639504/helping-minority-students-after-
brown-vs-board-ed.
Siwatu, K., Polydore, C., & Starker, T. (2009). Prospective elementary school teachers’
culturally responsive teaching self‐efficacy beliefs. Multicultural Learning and Teaching,
4(1), 1‐15. doi:10.2202/2161-2412.1040
111
Siwatu, K. O., & Starker, T. V. (2014). Preparing culturally responsive teachers. In G. S.
Goodman (Ed.), Educational psychology reader: The art and science of how people learn
(Rev. ed., pp.192-202). New York: Peter Lang.
Smith, P., & Bell, L. (2014). Leading schools in challenging circumstances: Strategies for
success. New York: Bloomsbury Academic.
Span, C. (2012). Reassessing the achievement gap: An intergenerational comparison of African
American student achievement before and after compensatory education and the
Elementary and Secondary Education Act. Teachers College Record, 114(6), 1-17.
Retrieved from http://www.tcrecord.org ID Number: 16690
Stecher, B., & Naftel, S. (2006). Implementing standards-based accountability (ISBA):
study design, state context, and accountability policies. Paper presented at
“Implementing No Child Left Behind: New Evidence from Three States” at the annual
meeting of the American Educational Research Association, San Francisco, CA.
Retrieved from
http://www.rand.org/pubs/working_papers/2006/RAND_WR380.pdf.
Sterbin, A. (2001). Rozelle Elementary School: A longitudinal analysis 1995-2000.
University of Memphis. Retrieved from www.mrsh.org/ipr.html
Stewart, B. (2006). Value-added modeling: The challenge of measuring educational
outcomes. New York: Carnegie Corporation of New York.
Stone, C. (1998). Leveling the playing field: An urban school system examines equity in access
to mathematics curriculum. Urban Review, 30(4), 295–307. Retrieved from
http://link.springer.com/article/10.1023%2FA%3A1023246618407#page-1
Stone, S., Engel, M., Nagaoka, J., & Roderick, M. (2005). Getting it the second time around:
112
Student classroom experience in Chicago's Summer Bridge Program. Teachers College
Record, 107, 935-957. Retrieved from http://www.tcrecord.org ID Number: 111845
Stonehill, R. M., Lauver, S. C., Donahue, T., Naftzger, N., McElvain, C. K., & Stephanidis, J.
(2011). From after-school to expanded learning: A decade of progress. New Directions
for Youth Development, 131, 29–41. doi:10.1002/yd.406.
Stronge, J., & Tucker, P. (2000). Teacher evaluation and student achievement.
Annapolis Junction, MD: NEA Professional Library.
Talbert-Johnson, C. (2004). Structural inequalities and the achievement gap in urban
schools. Education and Urban Society, 37(1), 22-36. doi:10.1177/0013124504268454
Tabachnick, B. G., & Fidell, L. S. (2012). Using multivariate statistics (6th ed.). Boston:
Pearson Education.
Takanishi, R. (2012). Compensatory education for all? Teachers College Record, 114(6), 1-6.
Retrieved from http://www.tcrecord.org ID Number: 16699
Tankersley, K. (2007). Tests that teach: Using standardized tests to improve instruction.
Alexandria, VA: Association for Supervision and Curriculum Development.
Thompson, M., Thompson, J., & Thompson, S. (2002). Catching kids up: Learning-focused
strategies for acceleration. Boone, NC: Learning Concepts, Inc.
Thum, Y. M. (2002). Measuring student and school progress with the California API.
National Center for Research on Evaluation, Standards, and Student Testing. Los
Angeles: University of California.
Thum, Y. M. (2003). No Child Left Behind: Methodological challenges & recommendations for
113
measuring appropriate yearly progress. CSE Tech Report 590, National Center for
Research on Evaluation, Standards, and Student Testing. Los Angeles: University of
California.
Tombar, T., & Borich, G. (1999). Authentic assessment in the classroom applications and
practice. Columbus, OH: Merrill.
Tomlinson, C.A. (1995a). Differentiating instruction for advanced learners in the mixed-ability
middle school classroom. ERIC Clearing House on Disabilities and Gifted Education.
Retrieved from ERIC (ED389141)
Tomlinson, C. (1995b). How to differentiate instruction in mixed-ability classrooms.
Alexandria, VA: Association for Supervision and Curriculum Development.
Tomlinson, C. A. (1999). The differentiated classroom: Responding to the need of all learners.
Alexandria, VA: Association for Supervision and Curriculum Development.
Trochim, W. (1984). Research design for program evaluation: The regression-discontinuity
approach. Beverly Hills: Sage Publications.
Tschannen-Moran, M., & Barr, M. (2004). Fostering student learning: The relationship of
collective teacher efficacy and student achievement. Leadership and Policy in
Schools, 3(3), 189-209.
Tucker, C. M., Porter, T., Reinke, W., Herman, K. C., Ivery, P., Mack, C., & Jack, E.
(2005). Promoting teacher efficacy for working with culturally diverse students.
Preventing School Failure, 50(2), 29-34. Retrieved from
http://www.researchgate.net/publication/254347125_Promoting_Teacher_Efficacy_for_
Working_With_Culturally_Diverse_Students
U.S. Department of Education. (2004). Four pillars of NCLB. Retrieved from
114
http://www.ed.gov/nclb/overview/intro/4pillars.html.
U.S. Department of Education, Institute of Education Sciences, National Center for Education
Statistics. (2005). NAEP Data Explorer. Retrieved from
https://nces.ed.gov/nationsreportcard/pdf/main2005/2006453.pdf
U.S. Department of Education. (2006). No Child Left Behind: The facts about math
achievement. Retrieved from http://www.ed.gov/nclb/methods/math/math.pdf).
U.S. Department of Education. (2007a). The condition of education. Retrieved from
www.ed.gov/about/pubs/intro/index.html?src=gu
U.S. Department of Education. (2007b). Free appropriate public education for students with
disabilities: Requirements under section 504 of the Rehabilitation Act of 1973. Retrieved
from http://www.ed.gov/about/offices/list/ocr/docs/edlite-FAPE504.htm
U.S. Department of Education. (2007c). IDEA regulations: Alignment with the No Child
Left Behind Act. Retrieved from
http://idea.ed.gov/explore/view/p/%2Croot%2Cdynamic%2CTopicalBrief%2C3
%2C
U.S. Department of Education. (2008a). The federal role in education.
Retrieved from http://www.ed.gov/about/overview/fed/role.html
U.S. Department of Education. (2008b). The final report of the National
Mathematics Advisory Panel. Retrieved from
http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf
U.S. Department of Education. (2009). Family Educational Rights and Privacy Act
(FERPA). Retrieved from http://www.ed.gov/policy/gen/guid/fpco/ferpa/index.html.
U.S. Department of Education (2011). Elementary and Secondary Education Act (ESEA).
115
Retrieved from http://www.ed.gov/esea
U.S. Department of Education, National Center for Education Statistics. (2011). The condition of
education, mathematics performance.Retrieved from
http://nces.ed.gov/fastfacts/display.asp?id=514
Urban, W., & Wagner, J. 2009. American education: A history. New York: Rutledge.
Valencia, R. (2012). Contextualizing “rethinking compensatory education”: The value of a
temporal continuity analysis. Teachers College Record, 114(6), 1-5. Retrieved from
http://www.tcrecord.org ID Number: 16689
Valero, P. (2012). A socio-political look at equity in the school organization of mathematics
education. In H. Forgasz & F. Rivera (Eds), Towards equity in mathematics education:
Gender, culture, and diversity (pp. 373-387). New York: Springer.
doi:10.1007/978-3-642-27702-3_34
VanDerHayden, A., & Burns, M. (2009). Performance indicators in math: Implications
for brief experimental analysis of academic performance. Journal of Behavior
Education, 18, 71-91. doi:10.1007/sl0864-0009-908-x
Ventura, J., 2014. The history of personalized learning. Milestones on the pathway to personalize
learning. Web log post via New Classrooms.
Retrieved from http://blog.newclassrooms.org/the-history-of-personalized-learning
http://blog.newclassrooms.org/the-history-of-personalized-learning
Vaughan, A. (2002). Standards, accountability, and the determination of school
success. The Education Forum, 66(3), 206-213. Retrieved in ERIC (EJ646599)
Viadero, D. (2006). U.S. pilot of AYP ‘growth’ models advances. Education Week, 25(31), 31.
Retrieved
116
from http://www.edweek.org/tm/contributors/debra.viadero.html
Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes.
Cambridge, MA: Harvard University Press.
Walston, J., & McCarroll, J. C. (2010). Eighth grade algebra: Findings from the eighth-grade
round of the Early Childhood Longitudinal Study, kindergarten class of 1998-99 (ECLS-
K, Statistics in Brief, NCES 2010-016). Washington, DC: National Center for Education
Statistics. Retrieved from https://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=201001
Waters, J. T., Marzano, R. J., & McNulty, B. A. (2003). Balanced leadership: What 30
years of research tells us about the effect of leadership on student achievement.
Aurora, CO: Mid-Continent Research for Education and Learning.
Webster, W. J. (1998). A comprehensive system for the evaluation of schools. Paper
presented at the annual meeting of the American Educational Research
Association, San Diego, CA.
Webster, W. J., & Mendro, R. L. (1995). Evaluation for improved school level decision making
and productivity. Studies in Education Evaluation, 21, 361-399.
Wehmeyer, M., Shogren, M., Palmer, S., Williams-Diehm, K., Little, T., & Boulton, A. (2012).
The impact of the self-determined learning model of instruction on student self-
determination. Exceptional Children, 78(2), 135-153. Retrieved in ERIC (EJ970668)
White, E. E. (1886). The elements of pedagogy. New York, NY: American Book.
White, E. E. (1888). Examinations and promotions. Education, 8, 517-522.
Wiggins, G. P., & McTighe, J. (2005). Understanding by design. Alexandria, VA: Association
for Supervision and Curriculum Development.
Wiles, J., Bondi, J., & Wiles, M. (2006). The essential middle school. New Jersey: Pearson.
117
Williams, J. (1996). How to manage your middle school classroom. Westminster, CA:
Teacher Created Resources.
Winters, M. A., & Greene, J. P. (2012). The medium-run effects of Florida's test-based
promotion policy. Education Finance and Policy, 7, 305-330.
doi:org/10.1162/EDFP_a_00069
Wormeli, R. (2006). Fair isn’t always equal assessing & grading in the differentiated classroom.
Portland, ME: Stenhouse Publishers.
Xia, N., & Kirby, S. N. (2009). Retaining students in grade: A literature review of the effects of
retention on students' academic and nonacademic outcomes. (Technical Report No. 678).
Santa Monica, CA. Retrieved from http://www.rand.org/pubs/technical_reports/TR678/
Yeh, S. (2010a). The cost effectiveness of 22 approaches for raising student achievement.
Journal of Education Finance, 36(1), 38-75. doi:10.1353/jef.0.0029
Yeh, S. (2010b). Understanding and addressing the achievement gap through individualized
instruction and formative assessment. Assessment in Education: Principles, Policy &
Practice, 17(2), 169-182. doi:10.1080/09695941003694466
Yell, M., & Walker, D. (2010). The legal basis of response to intervention: Analysis and
implications. Exceptionality, 18(3), 124-137. doi:10.1080/09362835.2010.491741
118
APPENDIX A SCHOOL APPROVAL
119
APPENDIX B IRB APPROVAL