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The comparison of hybrid intervention and traditional intervention in increasing student achievement in middle school mathematics

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The comparison of hybrid intervention and traditional intervention in increasing student achievement in middle school mathematics

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Content COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 1

THE COMPARISON OF HYBRID INTERVENTION AND TRADITIONAL
INTERVENTION IN INCREASING STUDENT ACHIEVEMENT
IN MIDDLE SCHOOL MATHEMATICS

by

Amber Terry McKinney
_______________________________________________________________

A Dissertation Proposal Presented to the
FACULTY OF THE USC ROSSIER SCHOOL OF EDUCATION
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF EDUCATION

December 2015

Copyright 2015 Amber Terry McKinney

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 2

Abstract
No Child Left Behind (NCLB, 2002), a reauthorization of the Elementary and Secondary
Education Act, mandates that districts and schools are accountability for student proficiency in
K-12 education. National, state, and local data has demonstrated an achievement gap in
mathematics. Institutions are consistently pursuing supplementary educational support programs
to support students at risk of not meeting proficiency. Effective intervention is a method to
improve student mathematics performance. The deficit fiscal climate has made efforts of
implementing effective intervention difficult. Institutions are faced with the task of providing
effective intervention with limited financial resources. Recently, institutions have considered
hybrid intervention programs that utilize both computer-assisted instruction and traditional
instruction. Hybrid intervention models employ a variety of research-based instructional
strategies and frequent assessment and feedback to improve mathematics achievement. This
study compared and evaluated the effectiveness of a hybrid intervention model and traditional
intervention model in a middle school by examining the differences in the mean scale scores of
the 2012 CST for Mathematics from a one-way between groups analysis of covariate
(ANCOVA) using the 2011 CST Mathematics scores as the control variable. The results from
the study suggested that the hybrid intervention had a significant impact on the student
performance. Further, a cost-effective analysis was conducted to determine the per-pupil cost to
increase student proficiency by one point on the CST for Mathematics. The data suggested that
the hybrid intervention is more cost-effective than the traditional intervention.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 3

DEDICATION
I dedicate my dissertation to my family. This would not have been possible without their
continued support, consistent motivation, and unwavering belief.
I have special feelings of gratitude to my Auntie Barbara for her endless and selfless
support; my children because they are my world and the focus of my life; my husband for
making me feel like “Superwoman” every day; my mother and father for always fostering my
self-efficacy and encouragement; and my sisters for patiently listening and letting me vent.
I also have esteemed appreciation for my great-grand parents Richard and Addie Jones
for instilling the importance of higher education and my granny, Sue Prudholm, for her moments
of wisdom. May they rest in peace.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 4

ACKNOWLEDGEMENTS
I would like to express my deepest gratitude to my advisor, Dr. Guilbert Hentschke for
his guidance, support, and patience. Without his push for me to finish and not give up, through
the hard times and unforeseen events, this dissertation would not have been possible. I would
like to thank my committee member, Dr. Helena Seli, for her encouragement and motivation.
For also, developing me professionally by engagement in innovative instructional technology. I
sincerely appreciate her for supporting me in my coursework during the birth of my daughter. I
would like to thank my committee member Dr. Dennis Hocevar for his support academically and
professionally. For also, building my conceptual understanding of statistics, which has
tremendously developed me as a professional.
I would like to also thank the coordinators and instructional coaches at my research site
for providing me the opportunity to perform my research and supporting me with my data
collection.
I would like to thank my colleagues, who are most importantly my friends for continually
encouraging me and supporting me.
Finally, I would like to thank my aunt, uncles, and cousins for being some of my best
cheerleaders.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 5

TABLE OF CONTENTS
List of Tables ...........................................................................................................7
List of Figures ..........................................................................................................8
Chapter One: Overview of the Study .......................................................................9
Background ..................................................................................................9
Mathematics Transition between the Primary and
Secondary Levels ...........................................................................10
Statement of the Problem ...........................................................................12
Purpose of the Study ..................................................................................13
Research Questions ....................................................................................14
Hypotheses .................................................................................................14
Summary of the Methods ...........................................................................15
Significance of the Study ...........................................................................16
Limitations, Assumptions, and Design Controls .......................................16
Definition of Key Terms ............................................................................17
Organization of the Study ..........................................................................18
Chapter Two: Review of Related Literature ..........................................................20
No Child Left Behind.................................................................................20
Common Themes in National, State-Wide, and Local
District Assessment Data ...............................................................22
Cost-Effectiveness Analysis ......................................................................24
Response to Intervention (RTI) .................................................................26
RTI in Mathematics ...................................................................................28
Elements of Effective Intervention ............................................................31
Hybrid Learning/Computer-Assisted versus Traditional Learning ...........36
Summary ....................................................................................................38
Chapter Three: Research Design and Methodology ..............................................41
Introduction ................................................................................................41
Research Questions and Hypotheses .........................................................42
Research Design.........................................................................................43
Data Collection and Instrumentation .........................................................48
Data Analysis .............................................................................................49
Summary ....................................................................................................51
Chapter Four: Analysis of Data .............................................................................52
Findings......................................................................................................52
Organization of Data Analysis ...................................................................52
Research Questions and Associated Hypotheses .......................................53
Analysis of Data .........................................................................................55
Summary ....................................................................................................74
Chapter Five: Findings, Conclusions, and Implications ........................................78
Summary of the Study ...............................................................................78
Findings of the Study .................................................................................81
Conclusions of the Study ...........................................................................82
Implications of the Study ...........................................................................86

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 6

Future Research .........................................................................................89
Summary ....................................................................................................91
References ..............................................................................................................93

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LIST OF TABLES

Table 1. Demographics of Population ...................................................................45
Table 2. Participant Treatment Sample Itemization ..............................................48
Table 3. Intervention Instructional Schedule .........................................................48
Table 4. Descriptive Statistics for Sixth-Grade 2012 Pretest
Mathematics CST Scores by Intervention ...............................................57
Table 5. Descriptive Statistics for the Algebra I 2011 Pretest
Mathematics CST Scores by Intervention ...............................................58
Table 6. Independent Sample t-Test for Sixth-Grade 2011 Mathematics .............59
Table 7. Independent Sample t-Test for Algebra I 2011 Mathematics ..................60
Table 8. Descriptive Statistics for the 2012 Mathematics
CST Scores by Intervention and Grade ..................................................61
Table 9. Test of Homogeneity of Slopes for Sixth-Grade Intervention
Groups .....................................................................................................64
Table 10. Levene’s Test of Equality of Variances for
Sixth-Grade Mathematic 2012 CST Scores ...........................................65
Table 11. Analysis of Covariance (ANCOVA) for Sixth-Grade
Mathematics ...........................................................................................66
Table 12. Test of Homogeneity of Slopes for Algebra I Intervention
Groups ...................................................................................................68
Table 13. Levene’s Test of Equality of Variance for
Algebra I Mathematics 2012 CST Scores .............................................69
Table 14. Analysis of Covariance (ANCOVA) for Algebra I Mathematics..........70
Table 15. Summary of Title I Expenditure Report by Intervention .......................72
Table 16. Average Per-Pupil Cost by Each Intervention Program ........................72
Table 17. Descriptive Differences by Intervention ................................................73
Table 18. Cost-Effectiveness Ratio by Intervention Group ...................................74
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LIST OF FIGURES

Figure 1 Sixth-Grade 2011 Average CST Scores by Intervention ..............................56
Figure 2 Algebra I 2011 Average CST Scores by Intervention ...................................57
Figure 3 Scatterplot of 2011 and 2012 Sixth-Grade CST Mathematics
Scores by Intervention ..................................................................................63
Figure 4 Scatterplot of 2011 and 2012 Algebra I CST Mathematics
Scores by Intervention ..................................................................................67

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CHAPTER ONE: OVERVIEW OF THE STUDY
This study examined middle school mathematics intervention programs, with the intent to
improve mathematics achievement at the lowest monetary expense. According the No Child
Left Behind (2002), all students were required to be proficient in mathematics by 2014. The
National Report Card for Mathematics in 2011 (NAEP, 2011) indicated that only 31% of
students in eighth grade were proficient or advanced in the state of California. Math proficiency
requires conceptual and procedural understanding (Kilpatrick, Swafford, & Findell, 2001). In an
effort to improve student proficiency, schools have implemented mathematics intervention.
Effective intervention requires the foundational mathematics knowledge and skills in addition to
extended learning time. This study examined and compared the cost-effectiveness between a
traditional intervention mathematics program and a hybrid intervention mathematics program.
Background
According to NCLB (2002), all K-12 students should have reached 100% proficiency in
mathematics by 2014. In an effort to make progress towards the 2014 goal, NCLB required
schools to meet the Academic Performance Index (API) and Academic Yearly Progress (AYP)
requirements. Schools that failed to meet the AYP goals faced sanctions that included
identification as a failing school, mandatory intervention, school restructuring, or school
takeovers (Mills, 2008).
The Nation’s Report Card (NAEP, 2011) indicated that students in the eighth grade
throughout the US are not meeting proficiency level in mathematics at all organizational levels.
The 2012 National Report Card indicated that as of the 2010-2011 school year, 39% of fourth
and 32% of eighth graders had met the goal while only 32% of graduating seniors were
proficient (Peterson, Woessman, Hanushek, and Lastra-Anadón, 2011, p. v). In addition, the
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 10

lack of achievement in mathematics was also demonstrated at the state and local levels as
reported on the California Standardized Testing and Reporting (STAR, California Department of
Education, 2011). Based on the data provided by the U.S. Department of Education (2011),
there was an average achievement gap of 44% between current student performance and the
AYP target. Most schools and districts have turned to intervention programs as a significant
method for providing additional support for struggling students and to increase student
proficiency in mathematics.
School districts across the country are failing to meet the target even though federal and
state funds were allocated specifically for the purpose of increasing mathematics achievement.
But the funds needed to sustain academic programs are decreasing due to fiscal deficits at the
national and state level, while the percentage of students required to meet the proficiency targets
has increased. School administrators continue to be expected to show optimal student
mathematics performance. This limited funding has a significant impact on decisions made by
stakeholders—educational agencies, administrators, school boards, educators, students, and
parents and tutoring companies, textbook publishers, software companies, and taxpayers.
According to Gardner (2007), because little to no funding is provided for supplemental
programs, administrators need to focus their attention on programs that produce the highest
achievement gain at the lowest cost. These programs are shown to be cost effective (Levin,
1995).
Mathematics Transition between the Primary and Secondary Levels
Many students entering middle school have a difficult time performing at grade level in
mathematics. According to Stewart (2010), these students lack proficient math skills because of
the difficult transition between primary and secondary levels. At the primary level, students
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 11

were, on average, in a self-contained class of 30 students with one teacher. The typical primary
system allowed teachers some leverage in modifying the structure of the curriculum based on the
needs of the students. Students that were having a difficult time understanding math concepts
had the advantage of receiving additional instructional time and increased small-group support
during class time. Primary students are taught more effectively when receiving more
individualized support (Stewart, 2010).
The secondary educational system was significantly different. At the secondary level,
students had a different teacher for each subject, and subject-specific classes were set at precise
times of the day with no latitude for modification. On average, students receive four to five
hours of dedicated math instruction per week. The unparalleled transition between the primary
and the secondary system accounted for part of the achievement gap (Stewart, 2010) and
demonstrated a need for additional mathematical support at the secondary level to provide more
flexibility and offer individualized support.
Mathematics assessment data showed a decrease in achievement between general
mathematics and Algebra I. In 2008, the California State Board of Education voted on a policy
that required all eighth-grade students to take Algebra I (Blume, 2008) regardless of their
demonstrated proficiency in seventh-grade mathematics. At the time the policy was approved,
only 21% of eighth-grade students in the current study enrolled in Algebra I had scored
proficient or higher on the CST (Blume, 2008). Proponents of the policy believed that eighth-
grade students taking Algebra I would force districts and schools to prepare students for eighth-
grade Algebra I in the preceding grades. According to Blume (2008), Critics believed that the
policy was unrealistic. Based on the National Report Card for 2011 (U.S. Department of
Education, 2012) and the STAR report for 2011 (California Department of Education, CDE,
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 12

2011), students are still not meeting the AYP benchmarks in Algebra I. The research also
indicated that students are not demonstrating proficiency in sixth- and seventh-grade
mathematics and, therefore, are not prepared for Algebra I in the eighth grade. According to
Ketterlin-Geller, Chard, & Fien (2008), the decline in mathematics achievement occurs at the
Algebra I level because students are unable to integrate and extend the skills they did not learn in
the primary years. To be successful, students need additional mathematics support outside of the
regular classroom day to develop the prerequisite skills needed to perform complex mathematical
operations that are the core of Algebra I.
Statement of the Problem
The NAEP (2011) mathematics data and the STAR data (CDE, 2011) indicated that there
is a significant achievement gap in mathematics. An approach to improving mathematics
achievement is by implementing effective mathematics intervention. Another key obstacle is
providing funding of supplemental intervention programs in a strained fiscal climate. The
problem that follows is determining a middle-school mathematics intervention program that
provides the optimal cost-benefit.
A student’s success in mathematics at the middle school level will have an impact on his
or her success in high school and beyond. According to Huetinck and Munshin (2004 as cited in
Hardré, 2011, p. 213), historically math is the subject that students find difficult and suffer
through the most. The NCLB (2002) mandates have created a high-stakes testing situation in
public schools. Schools are making the effort to meet the goals in mathematics through
intervention programs, school policies, teacher evaluations, and professional development. And
while some research has shown that students are gaining in proficiency in mathematics, the
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 13

increase was not sufficient enough to meet the AYP target of 100% proficiency in mathematics
by 2014 (NAEP, 2011).
In an effort to increase the proficiency of students in mathematics, schools have
implemented mathematics intervention. Research shows that math achievement can be increased
by providing research-proven programs (Slavin, Lake, & Groff, 2009). Effective research-
proven programs emphasized cooperative learning. Due to the current budget constraints, the
goal of schools is to provide effective intervention programs that can service as many students as
possible with limited funding. Computer-assisted programs increase the number of students that
can be provided intervention; however, computer-assisted programs do not work for all students.
Student success in a learning environment is based on their motivational orientation such as self-
efficacy, mastery orientation, and self-regulation (Clayton, Blumberg, and Auld, 2010).
Therefore, some schools offer both traditional and hybrid models. Since both models have
successful components, a hybrid model may be best in providing intervention services to K-12
students. Therefore, an attempt is being made to evaluate a real-world educational setting where
a hybrid and traditional intervention program are being utilized to provide support services for
middle-school students in increasing mathematics achievement with the highest efficiency.
Purpose of the Study
The purpose of the current study was to compare and evaluate the effectiveness of a
hybrid intervention program and a traditional intervention program in a middle-school setting for
increasing mathematics achievement. A quantitative methods design was applied to compare the
programs. This study explored which intervention program had the highest student achievement
gain based on the per-pupil cost.
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 14

Research Questions
1. Is there a significant difference in the performance level of students enrolled in the
traditional intervention programs and hybrid intervention programs?
2. Using prior grade CST scores as a control variable for previous math achievement, is
there a statistically significant difference in means scores on the 2012 CST for
Mathematics scores between students in the traditional intervention and hybrid
intervention?
3. Based on a cost-effectiveness analysis, which program will be most cost-effective
between the traditional intervention and hybrid intervention programs as reflected in the
per-pupil cost to increase one point on the CST for Mathematics?
Hypotheses
1. The data should not reflect a significant difference in prior year performance levels
between the traditional and intervention program. Sixth-grade students are automatically
enrolled in the program for one semester. For the purpose of this study only sixth-grade
students with CST achievement levels of Basic or lower were used as participants to have
matched achievement levels for this study. The data should not reflect a significant
difference in prior-year performance levels between the traditional and intervention
programs in Algebra I because the selection and enrollment process was parallel.
Students were selected for the intervention programs by the grade-level counselors,
instructional coaches and coordinators, and teacher recommendations. Prior year CST
data was used for primary recommendation. Students with CST Mathematics
achievement levels of Basic or lower were enrolled in the programs.
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 15

2. The CST data will demonstrate a significant mean difference in scores between the
traditional and hybrid intervention model, with the hybrid model showing a higher
significant effect size. Traditional intervention programs are comprised of direct
interaction with instruction that is primarily focused in whole-group instruction. Hybrid
intervention programs utilize a combination of direct and computer-assisted instruction
that is individualized to student needs and have shown to be effective in increasing
student performance.
3. There is limited research on the cost-effectiveness of supplementary educational
programs because of the novelty of computer-assisted programs, therefore, a reasonable
hypothesis could not be supposed. The observed data will assist in developing a post-
study hypothesis. This hypothesis will be developed based on the student achievement
gain on the CST and the per-pupil cost for each intervention program.
Summary of the Methods
The selection process for both programs targeted students that were below proficient in
mathematics. An Independent Samples T-test will determine if there is a significant mean
difference and effect size in CST Mathematics scores between the students enrolled in the
traditional intervention program and the hybrid intervention program, prior to starting the
intervention programs. The primary goal of the intervention programs was to increase student
achievement on the mathematics CST. An ANCOVA was conducted to determine if there was a
significant difference and the effect size between the CST scores of the students enrolled in the
traditional intervention and the hybrid intervention. The cost effectiveness of each program was
determined by conducting a cost-effectiveness analysis. The cost-effectiveness ratio determined
the per-pupil cost to increase one-point on the CST for Mathematics.
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 16

Significance of the Study
This study compared the cost-effectiveness of two middle school mathematics
intervention programs. The intervention programs examined in the study were a traditional
intervention model and a hybrid intervention model. The traditional intervention model
completed direct instruction and the hybrid intervention model consisted of computer assisted
instruction and direct instruction. The 2012 mathematics CST data was analyzed to determine if
the interventions had a significant effect on the students’ mathematics achievement. In addition,
a cost-effectiveness analysis was completed to determine the per-pupil cost to increase one point
on the mathematics CST.
Schools are presented with intervention programs that claim to be successful. District
administrators and school site administrators are given the responsibility of selecting the
programs that will produce the highest gain within the budget allotted. The demands of NCLB
(2002) on mathematics achievement, and the size of budget for supplemental programs are
significant factors for administrators to consider utilizing the cost-effectiveness analysis in
selecting and allocating funds for intervention programs (Hummel-Rossi & Ashdown, 2002).
This study serves as an example of how to use the cost-effectiveness analysis to determine if
mathematics intervention programs utilized in middle schools are efficient in increasing student
achievement.
Limitations, Assumptions, and Design Controls
This study used descriptive statistics and analysis of covariance (ANCOVA) to compare
the effectiveness of a traditional math intervention and a hybrid math intervention program. The
study had some limitations. The study was limited in scale. The study only included one large
urban middle school in the district. A significant population of the school was identified as low-
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 17

socioeconomic status and Latino. The study was limited to the number of students enrolled in
each program. The study did not represent all grade levels from the population. The study was
not truly randomized because the students were assigned to the intervention program based on
performance level from the assessment data. The study only used one traditional model program
that had approximately two different instructors for each grade level for sixth and eighth grade.
In addition, only one type of hybrid model program was used for sixth-grade and Algebra I
students. The school did not offer a hybrid model for seventh grade so a comparison for seventh
grade could not be done.
Definition of Key Terms
The following definitions of key terms are used to examine the research questions
presented in this study:
 Academic Yearly Progress (AYP) calculations: AYP calculations are “a series of
minimum competency performance targets defined by state education agencies that must
be met by schools and school districts to avoid sanctions of increasing severity”
(Springer, 2007, p. 1)
 Achievement or Performance levels: Achievement and performance levels are thresholds
based on a state’s assessments (Dee & Jacob, 2011). For the purpose of this study,
achievement will be based on the performance bands adopted by the State of California
as demonstrated on the California Standards Test (CST). These bands include Advance,
Proficient, Basic, Below Basic, and Far Below Basic
 Achievement Gap: The achievement gap reflects the disparity in academic performance
between groups of students (Person & Chou, 2009)
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 18

 Proficient: Proficient is the level of performance that demonstrates a student’s ability to
score above a specific performance threshold on state assessments based on the state’s
specific achievement standards (Hemelt, 2011). For the purpose of this study, Proficient
level or proficiency will be based on the threshold level designated by the State of
California as demonstrated on the California Standards Test (CST).
Organization of the Study
There is a significant mathematics achievement gap between the current performance of
the students and the academic targets set by of No Child Left Behind (2002). The
implementation of NCLB (2002) intensified focus on school-level accountability by establishing
performance standards for students, schools, and districts (Hemelt, 2011). Districts and schools
are not meeting the accountability standards set by NCLB. As students transition from the
primary to the secondary levels, the achievement gap expands. The transition causes a lack of
continuity in the systems operations and mathematics instruction. Students at the secondary
level do not receive the same individualized support available at the primary level and primary
students are expected to use foundational mathematical skills learned at the primary level to
complete the more complex mathematical tasks required at the secondary level.
The federal and state governments have provided schools with additional funds to support
supplementary educational programs used to foster student achievement in mathematics. The
current fiscal deficits have made funding supplementary programs difficult at best. School
administrators have to service an increasing number of students with limited resources.
Administrators have to implement programs that will produce the highest student gain at the
lowest cost.
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 19

The following chapters in this study consist of the Review of Related Literature,
Research Design and Methodology, Analysis of Data, and Findings, Conclusions, and
Implications. The Literature Review (Chapter 2) includes an analysis of other studies related to
the trends in the mathematics performance of students. In addition, the literature review explores
intervention efforts including Response to Intervention (RTI Action Network, n.d.), elements of
effective intervention, and the traditional and hybrid intervention models. Research Design and
Methodology (Chapter 3) is an investigation and comparison of the effectiveness of a traditional
math intervention program and a hybrid math intervention program in an effort to answer the
research questions. Research Design and Methodology provides an in-depth discussion on the
research design, data collection, and data analysis. Analysis of Data (Chapter 4) will examine
and provide a comprehensive description of the results from the study. Chapter 5, Findings,
Conclusions, and Implications, is a detailed explanation of the findings, implications for the
study, and recommendations for future research.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 20

CHAPTER TWO: REVIEW OF RELATED LITERATURE
The purpose of this study was to compare the effectiveness of traditional mathematics
intervention programs and hybrid mathematics intervention at the middle school level. For the
purpose of this study, effectiveness was measured based on the 2012 mathematics CST mean
scores. This literature review will provide an overview of the No Child Left Behind Act of 2001
(NCLB, 2002) and will focus on the common themes in national, state, and district data that
demonstrates the proficiency of students in mathematics. Next, the literature review includes an
investigation of secondary math and effective math instruction. In addition, the literature review
will compare the theory, purpose, and functionality of computer-assisted instruction and/or
hybrid learning. This review will explore Response to Intervention (RTI Action Network, n.d.),
assessment and elements of effective intervention, and the effectiveness of math intervention
programs including traditional programs and hybrid programs. Finally, this review will develop
understanding of a cost-effectiveness analysis and examine the usefulness for decision making.
No Child Left Behind (NCLB)
No Child left Behind Act of 2001 (NCLB, 2002) is the reauthorization of the nation’s
omnibus Elementary and Secondary Education Act of 1965 (ESEA, Springer, 2007, p. 556). The
goal of NCLB was for 100% of students to be proficient or higher in mathematics and English
Language Arts (ELA) by the 2013-2014 academic year. The purpose of the law was to close the
achievement gap and equalize education for all students by providing the necessary resources to
increase academic achievement. In addition to meeting the 2013-2014 100% goal of math
achievement in mathematics and ELA, schools were to meet state yearly goals, known as the
Academic Yearly Progress (AYP). The NCLB accountability systems require annual testing of
students in public schools in reading and math from third to eighth grade and once in 10th
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 21

through 12th grades (Dee & Jacobs, 2011). Schools that failed to meet the AYP goals each year
became program improvement (P1) schools. Schools that are identified as PI schools can face
consequences that include student choice transfers, supplemental intervention services, staff
replacement, curriculum management, outside experts, reopening as a charter school, and/or
school restructuring (Gardner, 2007).
The performance of the students for AYP goals are based on the state assessment. In
California, the AYP is based on the California Standardized Test (CST). With each state having
a different set of standards and assessments, many critics believe that the accountability of
NCLB (2002) is not uniformed or standardized. According to Gardner (2007), “Accountability
now depends on which subgroups are included in the system, how each state calculates AYP,
and which district, school, or subgroup benefits from the various changes states adopted” (p. 4).
Because the NCLB (2002) was based on mathematics and ELA scores, resources at many
school sites have been focused on those subjects. According to Dee & Jacob (2011) the
accountability policies of NCLB has caused educators to shift resources to the tested subjects,
mathematics and ELA, and shift away from subjects like history and science. Since mathematics
is a primary subject requirement for NCLB, schools that are not meeting the requirements are
required to include intervention. Mathematics intervention is vital to increasing the proficiency
of students in mathematics. Studies have suggested that improvements in student performance
can be attributed to the change in school accountability by NCLB that includes supplemental
educational services (Hemelt, 2011).
Title I funding was provided to school sites from the federal government and directly
fixed to NCLB (2002). The purpose of Title I was to ensure that all children have a fair, equal,
and significant opportunity to obtain a high-quality education and reach, at a minimum,
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 22

proficiency on challenging state academic achievement standards and state academic
assessments (p. 1). According to the Department of Education (2004), the purpose of Title I,
funding was to “improve[ing] the academic achievement of the disadvantaged” (Section 101).
Title I funding is given to schools that have a large percentage of low-socioeconomic students to
provide resources and supplemental services to increase student proficiency (Gardner, 2007).
Most school intervention programs are implemented using Title I funds. According to
Springer (2007), schools that are not meeting the AYP goals are more productive in improving
the proficiency of low performing students, because of the accountability of NCLB (2002)
including supplemental services.
Common Themes in National, State-Wide,
and Local District Assessment Data
The Nation’s Report Card
According to the Nation’s Report Card for Mathematics (NAEP, 2011), on average
eighth-grade students in California performed similar to the national average. The data also
showed an achievement gap at all organizational levels and a continual achievement gap for
minority and low-socioeconomic students. Similar data was reflected in local district assessment
scores. On average, students of low-socioeconomic status scored lower than students of higher
socioeconomic status. The data suggested that continued resources are needed to close the
achievement gap in mathematics for all students.
The Nation’s Report Card showed 43% on average of eighth-grade students in the nation
are proficient or advanced in mathematics. The data also demonstrated a 32% achievement gap
for minority students and a 39% achievement gap for low-socioeconomic students. According to
the Report, 31% on average of eighth-grade students were proficient or advanced in mathematics
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 23

in California. The data showed a decrease in math achievement between the national and state
level. In addition, the data showed an average achievement gap of 47% for minority students in
mathematics.
For the purpose of this paper, the District Assessment Data for Los Angeles Unified
School District will be represented because Los Angeles Unified School district is the largest
district in the state of California. Based on the National Assessment of Educational Progress
(NAEP, 2011), 19% of eighth-grade students scored proficient or advance in mathematics. At
the district level there was a 50% average achievement gap for minority students and a 26%
achievement gap for low-socioeconomic students. The data demonstrated that on average, most
students are not proficient in mathematics at the secondary level. According to the Nation’s
Report Card, “students performing at the proficient level, should apply mathematical concepts
and procedures consistently to complex problems in the five NAEP content areas” (NAEP, 2011,
p. 52).
The data from the Nation’s Report Card in 2011 indicated that most students are
performing at a Basic level. According to the NAEP (2011), “students performing at the basic
level should exhibit evidence of conceptual and procedural understanding in the five NAEP
content areas. This level of proficiency signifies an understanding of arithmetic operations–
including estimation–on whole numbers, decimals, fractions, and percents” (p. 52). The data
suggested that additional support is needed to support students in mathematics.
California Standardized Testing and Reporting (STAR) Test Results
Students take the California Standardized Test (CST) in mathematics annually. The CST
in mathematics is given to public school students in the state of California (California
Department of Education, 2011). The tests are used to assess students’ proficiency levels based
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 24

on the California State Content Standards in mathematics. The California Standards identify the
knowledge and the skills students should have at each grade level K-12. Students that scored
proficiency on the CST are meeting the grade level, State Content Standards. Based on NCLB
(2002), by the 2011 school year, 79% of the students should have scored proficient or higher in
mathematics.
The trend in data based on the NAEP (2011) is similar to the trend in data reported by the
STAR in California. Based on the STAR (2011) results, 53% of students in the sixth-grade and
50% of students in the seventh grade were proficient or advanced in mathematics. This
demonstrated an achievement gap between 26% and 29%. In addition, 47% of eighth-grade
students enrolled in Algebra I scored proficient or advance. This result indicates a 32%
achievement gap. There was a decline in proficiency between the state and district levels. Based
on the STAR (2011) results, 41% of sixth graders, 36% of seventh graders, and 32% of eighth
graders enrolled in Algebra I scored proficient or advance in mathematics. The results showed
an additional decline in proficiency between grades six and eighth in Algebra I. The results from
the STAR (2011) report suggested that additional support is needed in mathematics as the
student transitions to higher grades.
Cost-Effectiveness Analysis
The current state budget has made schools’ efforts difficult to meet the AYP targets set
by NCLB (2002). In an effort to increase student proficiency in mathematics within the current
fiscal restraints, school districts have focused their efforts on implementing intervention
programs that can increase student achievement with the lowest cost. According to Ashdown
and Humme-Rossi (2002), policy makers and school administrators are fiscally responsible for
the education programs that are administered and educators are interested in the effectiveness of
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 25

the programs on student performance; therefore cost and effectiveness are high priorities. Both
school districts and schools are presented with a variety of intervention programs that have
demonstrated success which makes it difficult to decide on an effective program with high cost-
effectiveness.
The most effective method for deciding on an intervention program is the cost-
effectiveness analysis (CEA). CEA can be used to compare and determine the most cost-
effective program based on the budget and educational needs of the school. CEA is appropriate
when the goal to reduce the number of students performing below proficiency is a primary
concern (Yeh, 2010) and should be utilized when making decisions. According to Levin (1995),
“Cost-effectiveness analysis is a decision-oriented tool that is designed to determine which
means of attaining a particular educational goal is most efficient (p. 381).
There are criteria that need to be defined prior to starting the cost analysis that include
choice problems, measurement of effectiveness, program alternatives, and effects of the program
(Levin, 1995). The programs used in a cost-effectiveness analysis must demonstrate evidence of
increased student achievement (Hummel-Rossi and Ashdown, 2002). This suggested that all
alternative programs that are considered must have independently shown effectiveness. In
addition, the overall cost of the interventions is needed. The overall cost of an intervention
includes all necessary resources to effectively execute the program that includes, but is not
limited to, salaries and educational resources such as books, software, site licenses, and
technology. Levin (1995) referred to these resources as ingredients and defines a systematic
approach of three phases to estimate cost of a program that include 1) identification of
ingredients, 2) determination of the cost of each ingredient and 3) overall cost of the
intervention, an analysis of the cost, the agency fiscally responsible, and the duration of the
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 26

program (Hummel-Rossi and Ashdown, 2002). To accurately conduct a cost analysis, it is
imperative that the costs of all “ingredients” are taken into account and the effectiveness of the
program.
The result of a cost-effectiveness analysis is represented in a ratio known as the “cost-
effectiveness ratio (Levin, 1995) that represents the cost of each program in dollars and each
program’s outcome based on the appropriate unit of measurement (Ashdown & Hummel-Rossi,
2002). The cost-effectiveness ratio of an intervention program would represent the overall cost
of the program and the increased proficiency based on the CST data. The most desirable
intervention program based on a cost analysis is the program that has the lowest cost per
achievement gain (Levin, 1995; Ashdown & Hummel-Rossi, 2002).
Response to Intervention (RTI)
The implementation of the No Child Left Behind Act of 2001 and the Individuals with
Disabilities Education Act of 2004 (IDEA, Building the Legacy: IDEA 2004, n.d.) has increased
schools’ accountability for student achievement. In an effort to meet the AYP goals mandated
by the NCLB (2002), many schools have implemented the Response to Intervention (RTI Action
Network, n.d.) model. According to Fuchs and Fuchs (2006), “RTI is a means of providing early
intervention to all children at risk for school failure (p. 93). The primary goal of the RTI model
is to assist all students in achieving at a proficiency level (Burns, 2008). The RTI model has two
acceptable approaches: problem solving and standard protocol.
According to Grimes (2002 as cited in Fuchs & Fuchs, 2006), the problem solving
approach is a systematic process that includes problem identification, analyze causes, designed
interventions, implemented intervention, progress monitoring, and modification. The purpose of
the process is to use data as the foundation for instruction and intervention. In addition, the
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 27

problem solving approach is differentiated for each student. The standard treatment protocol is
not differentiated. According to Fuchs and Fuchs (2006), the standard treatment protocol is
implemented in fixed durations and instruction is provided in small groups.
The key difference between the two protocols is the problem solving model meets the
individual needs of multiple students, while the standard treatment protocol meets the needs of a
homogenous group (Lembke, Hampton, & Beyers, 2012). Researchers suggested that the
standard treatment protocol method is most effective because it requires all instruction to be
standardized and using one protocol ensures implementation fidelity; however, more school
districts have adopted the problem-solving approach that requires teachers to be experts in
assessment and intervention methods (Fuchs & Fuchs, 2006). Utilizing data to make informed
decisions is the most effective method to increase student proficiency. Programs that are
individualized to student needs foster student learning.
The RTI model is a multi-tier intervention model. The first tier is the general classroom
instruction (Johnson & Smith, 2008). The purpose of Tier 1 is to ensure that students have
equitable learning opportunities through effective pedagogy and data-driven instruction. During
Tier I level of RTI, teachers are observed for best instructional practices and student achievement
is monitored to determine if possible intervention is needed. Teachers are expected to use
research-based instructional strategies with evidence of student learning. Screening of students
that need intervention is initiated by the classroom teachers.. Teachers monitor student
achievement using formative and summative assessments. A primary method of screening
utilized by teachers is the CST and periodic assessment data from the prior year. CST data and
district assessment data are used to determine if the district assessment scores and CST scores
are parallel and to identify the discrepancy in specific concepts and skills.
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 28

Tier 2 is the intervention process for students who fail to meet the academic standards.
Johnson & Smith (2008) state, “This level is designed for students who do not achieve to grade-
level standards when provided with generally effective Tier 1 instruction.” (p. 49). Intervention
at this level can be implemented in multiple facets including small group instruction,
individualized tutoring, intervention electives, and computer-assisted instruction. Students
participate in intervention in addition to receiving regular classroom instruction. During the Tier
2 level, it is vital that progress monitoring is occurring to determine if students are improving
and meeting the academic standards. According to Ciullo, SoRelle, Kim, Seo, and Bryant
(2011), “educators must have access to progress-monitoring techniques that can be easily and
reliably used to identify students in need of intervention and to determine their response to
intensive instruction (p. 121).
Students that respond to the intervention and meet the academic standards move back to
Tier 1 and students who do not respond to Tier 2 level will move to Tier 3. Tier 3 is usually the
initiation process for special education. The research suggested that RTI is an effective model
for improving student achievement through effective pedagogy, intensive instruction, and
progress monitoring.
RTI in Mathematics
RTI has been used mostly in reading. According to Lembke et al. (2012), the core
principles of RTI reading research can be used in a mathematics model. The six core principles
that can be used for mathematics are: 1) All students can learn with effective instruction,
2) universal screening throughout the academic year, 3) progress monitoring, 4) research-based
instruction in the core class and intervention, 5) tiered instructional support and trained
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 29

educators, and 6) ongoing program evaluation. Implementation of the core principles can foster
student achievement in mathematics.
In addition to the six core principles, Lembke et al. (2012) described key elements that
are significant in an RTI model for mathematics. The first key element is utilization of a reliable
screening tool. The screening tool is used to identify students of concern. The authors suggested
that the screening tool should be used at least three times per year—fall, winter, and spring. This
provides the opportunity to identify students of concern throughout the year. In most districts,
the screening tool used is the benchmark or periodic assessments (Burns, 2008). In addition to
student identification for intervention, the benchmark assessments are also used to guide
instruction and intervention within the regular classroom. The authors suggested that the
screening measures that should be employed are based on the Curriculum-Based Measurement
(CBM).
Based on CBM, the universal screening for mathematics intervention at the secondary
level is estimation and Algebra I (Burns, 2008). The second element is core instruction and
intervention must be supported by research evidence and implemented with fidelity. An
effective way to determine if students are grasping the mathematical concepts and skills is
progress monitoring. Progress monitoring is essential to identifying student response to
instruction (Ciullo, et al., 2011). The final key element is data-based decision making. Data can
be monitored by the individual teacher, subject-grade-level teams, and cross-curricular teams
(Burns, 2008). The purpose of data-based decision making is to determine if the instruction or
intervention is effective for increasing student achievement and identifying skill deficits. Ciullo
et al. (2011) stated, “Assessment plays an integral role in helping teachers to identify
mathematical domains in which students require intensive intervention” (p. 120).
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RTI Tier 1 mathematics instruction should be accessible to all students and follow the
state standards (Lembke et al., 2012). The authors suggested that effective mathematics
instruction at the Tier 1 level should include differentiated instruction, cooperative learning, and
teacher self-checks. The purpose of Tier 1 is to create and modify lessons as needed to increase
student achievement. Lembke et al. (2012) suggested that the Tier 2 intervention (for students
who have reached proficiency) should be four to five times per week, for 20-40 minutes in
addition to 10-15 weeks of explicit teacher-led instruction. Tier 2 should be an extension of the
regular math instruction that provides students with additional time to develop key skills and
concepts. According to Griffiths, VanDerHeyden, Parson, and Burns’ (2006) study on reading
RTI, the most effective intervention is ten weeks consisting of 50 sessions of 35 minutes. In
addition, Tier 2 instruction should include targeted math skill instruction, instructional models,
guided practice, feedback, and repetition. This level of instruction is designed to provide the
additional support and foundational skills needed to increase understanding. Along with
instruction, progress monitoring should be used to ensure that students are making adequate
process.
According to Ciullo et al. (2011), progress monitoring tools can include daily checks, unit
checks, and aim checks. Daily checks are used to determine if students in intervention have met
the lesson objective, unit checks determine if students in intervention have mastered the unit
contents, and aim checks determine if the students are making progress towards their
intervention goals (Ciullo, et al., 2011). Varieties of progress monitoring tools allow for rapid
feedback and enable formative assessments to be completed several times per week. This form
of progress monitoring is similar to Rapid Assessment Systems (RSS). RSS provides student
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performance data and feedback two to five times per week which has been shown to increase
student engagement and achievement (Yeh, 2008).
RTI requires schools to use data-driven instruction and progress monitoring to increase
the achievement of students performing below grade level. RTI has led to higher gains in
student achievement (Burns, 2008).
Elements of Effective Intervention
The primary concern of intervention is the overall effectiveness as demonstrated by
student performance. Intervention is only effective if students show progress towards
proficiency. Effective mathematical intervention should operate as a corrective function, should
provide students with foundational mathematics knowledge and skills, and extended time for
learning standards-based mathematics content (Ketterlin-Geller et al., 2008). Mathematics
intervention in any capacity should be an extension of the math contents that are taught in the
regular math class. Effective interventions include the following key elements 1) peer-to-peer
instructional skill practice with multiple opportunities for rapid responses and immediate and
corrective feedback, 2) independent time practice, 3) slightly delayed corrective feedback, and
4) leveled progression based on the mastery of each skill level (VanDerHeyden & Burns, 2005).
Progress monitoring and effective instructional strategies are the foundation of a successful
intervention program.
Curriculum-Based Assessment and Curriculum-Based Measurement
A critical component of instruction is based on assessment data. Valid assessment data
are critical components of mathematics curriculum, effective instruction, and intervention
planning (VanDerHeyden & Burns, 2005, 2008). Data-driven instruction is the most effective
way to make sure that instruction aligns with students’ instructional needs (Kellterlin-Geller,
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 32

Chard, & Fien, 2008). Administrators and educators utilize state and district data for effective
lesson planning and lesson delivery. With the mandates of NCLB (2002) to meet the AYP
targets and the implementation of RTI as a means to increase student proficiency, schools need
effective methods of assessing and measuring student progress. According to VanDerHeyden &
Burns (2008), there are three areas of mathematics curriculum that should be examined that
include, what and when to measure, the response measurement or levels of proficiency, and
relevant standards that demonstrate math proficiency. Screening and progress monitoring are
essential components for effective math instruction and intervention.
Curriculum-Based Assessment (CBA) and Curriculum-Based Measurement (CBM) has
been shown to provide a comprehensive and instructionally relevant assessment model that
utilizes progress monitoring (VanDerHeyden & Burns, 2005). In addition, the authors stated that
CBA is used to assess the instructional needs of students based on ongoing performance using
the current academic content and standards. The CBA and CBM model is an effective
assessment tool and is used as a lens to determine if the intervention is effective based on student
performance. CBA and CBM can be used to evaluate the effectiveness of an intervention in
multiple ways including district assessments, state assessments, and fluency data. In addition,
CBA works in a sequential system, the performance of each skill is measured for mastery. CBM
is used to measure the progress towards proficiency (VanDerHeyden & Burns, 2008). The CBA
and CBM models mimic a formative and summative assessment model; however unlike
formative and summative assessments, CBM allows for often needed instructional adjustments
(Fore, Burke, Boon, & Martin, 2007).
Formative assessments are used to determine the specific objective and skills that need to
be taught and summative assessments are used to assess the learning or progress (Burns, 2011).
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 33

In some current instructional settings, the frequency of summative assessments do not occur
often enough to allow for instructional adjustments or too late to reduce student failure and
increase student proficiency. The CBA and CBM models can be an effective intervention model.
CBA can be used to determine the level of instruction and instructional strategies needed for the
intervention and CBM can be used to evaluate the overall effectiveness of the intervention based
on desired outcome (VanDerHeyden & Burns, 2005). The desired outcome of mastery in most
school settings is to score proficient on district assessment as a progress monitoring
measurement and the state assessment as a summative assessment measurement. Educators that
have used the CBA and CBM models have demonstrated significantly greater progress then
educators that used other methods of instruction and intervention (Fore et al., 2007).
According to Fore et al. (2007), the CBM model should be used as part of a problem-
solving approach that includes problem identification, problem analysis, intervention plan
development, intervention plan implementation, and intervention plan evaluation. This five step
CBM problem-solving approach is similar to the six stages of the RTI model suggested by
Lembke et al. (2012). In the problem identification stage, at-risk (students performing below
proficient are considered to be at risk) students for mathematics failure are identified using
district assessment scores, CST scores, and academic grades. In the problem analysis stage, a
hypothesis is identified based on the performance of the students. Educators, intervention
coordinators, and instructional coaches analyze student cumulative files and transcripts to
determine possible reasons for low-student performance. Cumulative files and transcripts
provide academic grades, assessment data, student behavior, student progress, language barriers,
physical disabilities, and academic disabilities. Classroom observations contribute to the data
collection as well. During the intervention plan stage, a program of action is developed that
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 34

includes acquisition and fluency level instruction. According to VanDerHeyden and Burns
(2005), acquisition-level instruction involves building fluency at a slightly lower level in order to
allow students to perform with accuracy and fluency. During the implementation stage,
intervention is designed to take place during the instructional setting. During the evaluation
stage, student progress is measured using CBM. This stage allows for educators to develop an
instructional plan that assists with student proficiency and makes modifications to the plan when
necessary. The evaluation stage also identifies if students are making adequate progress (Fore et
al., 2007).
The use of CBA and CBM in a problem-solving approach has been proven to increase
mathematics scores (VanDerHeyden & Burns, 2005). This model provides a systematic method
that allows for a comprehensive approach that produces data-driven decision making of effective
instructional strategies and intervention at the RTI Tier 1 and Tier 2 levels.
Effective Instructional Strategies
The key elements of effective intervention in a regular classroom setting uses specific
instructional strategies at multiple performance levels that include the acquisition level of
performance and fluency level of performance. According to VanDerHeyden and Burns (2008),
acquisition level strategies include modeling, guided practice, and frequent feedback. At this
level students show slow, but accurate, proficiency.
In the instructional level, students are expected to make errors while learning new skills.
Instructional strategies involve independent timed practice, increased student self-efficacy, and
multiple methods to demonstrate proficiency (VanDerHeyden & Burns, 2008). Effective
intervention should include instructional strategies that increase the fluency of a specific concept
or skill. A meta-analyses conducted by the researchers at the Mid-Continent Research for
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 35

Education and Learning (McREL) identified effective instructional strategies presented by
Marzano, Pickering, and Pollock (2001 as cited in Hoover, 2006) that have a positive effect on
increased fluency and student achievement. Marzano et al. (2001 as cited in Hoover, 2006)
strategies include: 1) Identifying similarities and differences; 2) summarizing and note taking;
3) Reinforcing effort and providing recognition; 4) Homework and practice; 5) Nonlinguistic
representation; 6) Cooperative learning; 7) Setting objectives and providing feedback;
8) Generating and testing hypothesis; and 9) Cues, questions, and advance organizers (p. 26). In
the fluency level, students are expected to perform at optimal fluency. At this level, students
demonstrate mastery of the skills and concepts. Instructional strategies that utilize acquisition
and fluency level performance will increase conceptual and procedural knowledge of
mathematical concepts and student proficiency.
Math proficiency is composed of both conceptual understanding and procedural
knowledge (Kilpatrick et al., 2001). Conceptual knowledge is the understanding of the
relationships that motivate math problems, and procedural knowledge is the understanding of the
rules and steps needed to solve the problems (Hiebert & Lefevre, 1986, p. 7). Effective math
interventions should incorporate both conceptual and procedural knowledge.
Conceptual knowledge of mathematics allows students to construct mathematical
knowledge in relevant and meaningful context (Ketterlin-Geller et al., 2008). According to
Hardré (2011),
Through exposure to multiple ways to engage and use math, their students can develop
linkages (cognitive schema) that cue recall and build even richer cognitive networks, to
improve math performance at school, as well as to transfer math distally, outside of
school and beyond their current experiences. (p. 230)
Procedural knowledge allows students to practice a sequence of logical steps and build mental
representations of the problem (Burns, 2011).
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Conceptual knowledge should be the first stage of intervention (Burns, 2011). It occurs
at the acquisition level of performance while procedural knowledge occurs at the fluency level of
performance. Students in need of mathematical intervention need to have a solid foundation to
develop their conceptual knowledge at the acquisition performance level. To develop the
procedural knowledge at the fluency level of performance, students need to accurately perform
mathematical operations; however, math proficiency goes beyond computations and operations.
Math proficiency is based on the ability to apply conceptual and procedural knowledge to solve
complex problems (VanDerHeyden & Burns, 2008).
The key to solving complex problems is to increase critical thinking skills. According to
Gasser (2011), there are five instructional strategies that can be incorporated in math instruction
to increase student proficiency. They are: “1) Incorporating problem-based instruction,
2) Fostering student-led solutions, 3) Encouraging risk taking, 4) Having fun, and 5) Providing
ample collaboration time for both our students and ourselves” (p. 109). Marzano et al.’s (2001
as cited in Hoover, 2006) strategies for effective instruction and Gasser’s (2011) strategies for
increasing critical thinking skills will foster student learning and enable students to develop a
comprehensive understanding of math concepts and skills to increase student achievement.
Hybrid Learning/Computer-Assisted versus Traditional Learning
Hybrid Learning or Computer-Assisted Learning is a combination of traditional
classroom instruction and computer-mediated instruction (Tuckman, 2001). Tuckman (2001)
described traditional classroom instruction as learning that takes place in a classroom with an
instructor and textbooks. Computer-mediated instruction is identified as learning that takes place
virtually and is acquired from performing with frequent feedback and progress monitoring rather
than listening. According to Sinclair, Renshaw, and Taylor (2003), “Computer-assisted
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 37

instruction has been shown to enhance rote memory skills and improve higher order critical
thinking skills” (Abstract).
According to Urtel (2008 as cited Diaz and Entonado, 2009) and Georgouli et al. (2008
as cited Diaz and Entonado, 2009), “the knowledge acquired by teachers who use online and
face-to-face methods can be of great use in improving both types of teaching (p. 331). As in all
approaches, hybrid learning has strengths and weaknesses depending on the model and type of
student.
Diaz and Entonado’s (2009) study evaluated traditional learning and online learning. The
results demonstrated that in the online model, the structure and theoretical content is more
effective than that of the traditional model. In the online model, students are consistently
developing schema through conceptual maps. In the traditional classroom, however, conceptual
development is often lacking. Diaz and Entonado also found that activities in the online model
are more effective than the traditional model because the activities are the fundamental element
of the program. In the traditional classroom, lecture is the foundation for learning; however, the
results indicated that the interaction between student and teacher is more effective in the
traditional classroom model because visual contact promotes student motivation.
Students’ motivational orientation towards learning has an effect on their preference of
learning environment (Clayton, et al., 2010). Clayton et al.’s (2010) study suggested, “that the
motivational factors achievement goals, self-efficacy, and learning strategies influenced student
choice in online, hybrid, or traditional learning environments” (p. 349). A student’s motivational
beliefs are linked directly to the preference of learning environment and learning strategies.
Students who receive instruction in their preferred environment will have higher academic
outcomes. The results from the study indicated that students who preferred the traditional
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 38

learning environment have a higher mastery goal orientation. Students who preferred online
learning environments have higher self-efficacy. Students who chose the hybrid model liked the
idea of having a combination of online and traditional learning strategies. The results from both
Diaz and Entonado (2009) and Clayton et al. (2010) suggested that hybrid learning is the most
effective learning environment.
Hybrid learning provides a variety of learning strategies that can meet the needs of all
students. The model allows for individualized support based on the needs of each student. An
effective learning model is designed to fit the needs, goals, talents, and interests of the learner
(Klasnja-Milićević, Vesin, Ivanović, and Budimac, 2011). In addition, the hybrid model offers a
learning structure that allows students to have direct classroom interaction and to exercise some
control over their learning. According to Clayton et al. (2010), “learners want engaging learning
environments that promote ‘direct interaction with professor(s) and students’, ‘spontaneity’,
‘immediate feedback’ and relationships with faculty and students” (p. 362). Tuckman’s (2001)
study suggested that students in a hybrid learning model show a significant academic increase.
He concluded that the hybrid model was more effective because students benefited from the
discipline provided by the traditional classroom instruction and the opportunities for practice,
assessment, and feedback provided by the hybrid model. The research suggested that hybrid
learning provides students with conceptual knowledge, theoretical knowledge, and procedural
knowledge that fosters student participation in the learning process. When students are active in
the learning process, they are more likely to develop the content skills and knowledge.
Summary
The goal of No Child Left Behind (2002) was to provide equitable educational
opportunities to all students but the evaluation of the literature suggested that an achievement
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 39

gap exists in the area of mathematics. Supplemental support is needed to close the achievement
gap. The evaluation of the literature revealed that students need mathematics intervention most
significantly at the secondary level. In an effort to service students at risk within the fiscal
deficit, the research suggested the use of the cost-effectiveness analysis to compare and identify
programs that produce the most student gain for the lowest cost. To provide supplemental
support for at-risk students, many school administrators adopted the Response to Intervention
(RTI) model, an early intervention, multi-tier system that uses data-driven instruction and
progress monitoring to increase student achievement.
RTI requires that a progress monitoring and evaluative tool be used to measure the
effectiveness of the intervention program. Marzano’s (2001 as cited in Hoover, 2006) strategies
for effective instruction and Gasser’s (2011) strategies for critical thinking have shown that using
data-driven strategies increases student performance. Curriculum-Based Assessment (CBA) and
Curriculum-Based Measurement (CBM), together, form an effective assessment tool that uses a
systematic, progress monitoring approach. The research shows that CBA and CBM have been
effective in data-driven decision making and shown greater progress in student achievement.
Hybrid programs are the newest intervention approach. School administrators have used
traditional intervention programs and hybrid intervention programs in an effort to foster student
achievement. Research has shown that hybrid interventions are able to provide a more
individualized approach that results in a higher progression for student achievement; however,
the effectiveness of the programs have not been measured in conjunction with the cost-
effectiveness analysis. Utilizing a cost-effectiveness analysis will isolate effective intervention
programs that achieve optimal student success at the lowest cost. The goal of this research was
to examine the effectiveness of traditional and hybrid intervention models based on assessment
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 40

data and the cost-effectiveness analysis. This study will assist stakeholders with identifying and
assessing math intervention programs that are academically and cost effective.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 41

CHAPTER THREE: RESEARCH DESIGN AND METHODOLOGY
Introduction
No Child Left Behind (NCLB (2002) forced schools to be accountable for the
achievement of the students. NCLB mandated 100% student proficiency or higher in
mathematics by 2014. Schools are required to meet Academic Yearly Progress (AYP) targets to
demonstrate steady progress toward the goal. Assessment data at the national, state, district, and
school site have demonstrated an achievement gap between student performance and the AYP
targets.
School districts have adopted the Response to Intervention (RT1) model to address the
achievement gap. RTI is effective because the model uses research-based strategies and progress
monitoring tools to increase student performance.
Government agencies provided funding for supplemental programs to assist with
increasing student achievement in mathematics; however, the current fiscal deficit has decreased
the funding at many schools. School administrators are given the task of implementing effective
intervention at the lowest cost; therefore, it is vital for them to identify effective, cost-beneficial
intervention.
The purpose of this study was to compare and determine the most cost-effective program
between a traditional mathematics intervention program and a hybrid intervention program. This
study used CST data and total cost estimation of the programs to compare and analyze the
programs. The research study design, participants, instrumentation, and data analysis to compare
the effectiveness of a hybrid model of intervention and traditional model of intervention in
mathematics for increasing student proficiency are discussed in this Chapter.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 42

Research Questions and Hypotheses
Research Questions
Following are the research questions that guided this study.
1. Is there a significant difference in the performance level of students enrolled in the
traditional intervention programs and students enrolled in hybrid intervention programs?
2. Using prior grade CST scores as a control variable for previous math achievement, is
there a statistically significant difference in mean scores on the 2012 CST for
mathematics scores between participants in traditional intervention and hybrid
intervention?
3. Based on a cost-effectiveness analysis, which program will be most cost-effective as
reflected in the cost per pupil to increase one point on the CST for Mathematics?
Hypothesis
Guiding this study, the following assumptions were used:
1. The data should not reflect a significant difference in prior year performance levels
between the traditional and intervention program. Sixth-grade participants are
automatically enrolled in the program for one semester. For the purpose of this study,
only sixth-grade students with CST achievement levels of Basic or lower will be used as
participants to matched achievement levels for this study. The data should not reflect a
significant difference in prior-year performance levels between the traditional and
intervention programs in Algebra I because the selection and enrollment process was
parallel. Participants were selected for the intervention programs by the grade level
counselors, instructional coaches and coordinators, and teacher recommendations. Prior
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 43

year CST data was used for primary recommendation. Participants with CST
Mathematics achievement levels of Basic or lower were enrolled in the programs.
2. The CST data will demonstrate a significant mean difference in scores between the
traditional and hybrid intervention model, with the hybrid model showing a higher
significant effect size. Traditional intervention programs are comprised of direct
interactions that are primarily focused in whole group instruction. Hybrid intervention
programs use a combination of directed and computer-assisted instruction that is
individualized to meet student needs and has shown to be effective in increasing student
performance.
3. There is limited research on the cost-effectiveness of supplementary educational
programs because of the novelty of computer-assisted programs; therefore a reasonable
hypothesis could not be supposed. The observed data will assist in developing a post-
study hypothesis that will be developed based on the student achievement gain on the
CST and the cost-per pupil for each program.
Research Design
This study investigated and compared the effectiveness of the traditional math
intervention program, After-School Academy for Success, the hybrid intervention program, and
the Math Tutorial Lab, on the math achievement of sixth-grade participants and eighth-grade
participants enrolled in Algebra I scoring below proficient on the CST for Mathematics. The
data was evaluated to determine if the mean of the 2011-2012 CST scores of students enrolled in
sixth-grade math or eighth-grade Algebra I were different between participants enrolled in
traditional intervention and those enrolled in hybrid intervention programs. The mean CST
mathematics scores of the participants enrolled in the After-School Academy of Success and the
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 44

Math Tutorial Lab were analyzed to determine if there was a significant statistical difference and
effect size between the outcomes of the two intervention programs.
The study also investigated the cost effectiveness of each program by calculating the
cost-effectiveness ratio. The cost-effectiveness ratio provided the per-pupil cost to increase one
point based on the average per-pupil cost. A cost-analysis was completed to determine the total
cost of the each program based on the resources needed to support the program including cost,
site license, materials, and staff positions. An average per-pupil cost was calculated, and the
cost-effectiveness ratio was calculated based on the per-pupil cost and mean difference in the
CST mathematics scores.
Study Setting
Sixth-grade Basic math and eighth-grade Algebra I participants who scored Basic, Below
Basic, or Far Below Basic on the CST mathematics test were enrolled in one of the intervention
models at a middle school in Los Angeles County. Participants in the hybrid intervention model
received 54 minutes of computer-assisted instruction in the Math Tutorial Lab three times a week
and 54 minutes plus 39 minutes of direct instruction twice a week. Participants enrolled in the
traditional intervention received 60 minutes of direct instruction twice a week. For participants
enrolled in the intervention programs, instruction was facilitated by a credentialed secondary
mathematics teacher. The participants also received 54 minutes (four days a week) and 39
minutes (one day per week) of regular mathematics instruction using the California State
Mathematics content standards. Data on student achievement was collected through the
administration of the CST for Mathematics.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 45

Population
The subjects of this study consisted of 632 sixth-grade, 748 seventh-grade, and 643
eighth-grade participants enrolled at a middle school in Los Angeles County. The study
examined CST scores in mathematics of May of 2012. The racial breakdown of the middle
school (grades 6 through 8) consisted of 0.3% Native Americans, 1.7% Asian, 4.0% Pacific
Islanders, 18.8% Filipinos, 62.3% Hispanic, 9.8% African Americans, and 3.3% Caucasians. Of
the middle school population, 79% of the students received free or reduced lunch. Table 1 shows
the demographics of the school’s population.

Table 1
Demographics of Population
Native
American

Asian
Pacific
Islander

Filipinos

Hispanic
African
American

Caucasian
Low
SES
0.3% 1.7% 4.0% 18.8% 62.3% 9.8% 3.3% 79.0%

Participant Selection
The selection process for this study was not random. Participants were selected for both
intervention programs based on their CST scores from the 2011 school year. Students scoring
below proficient were the targeted group for participation in the study. Students scoring Basic,
Below Basic, and Far Below Basic demonstrated an achievement gap in mathematics and were
enrolled in one of two intervention programs: (a) the traditional intervention program based on
100% teacher directed instruction for 120 minutes per week, or (b) a hybrid intervention program
that was comprised of 37% teacher-directed instruction, 93 minutes per week and 63% of
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 46

computer-based instruction, 162 minutes per week. All sixth-grade participants were enrolled
into the hybrid intervention program for one semester. For the purpose of this study, only sixth-
grade participants performing Basic, Below Basic, or Far Below Basic were used as participants
to match achievement levels in both programs. The Algebra I eighth-grade participants were
assigned to the hybrid intervention model by the guidance counselors and instructional coaches
based on their CST achievement level of Basic or lower. Students were assigned to the
traditional intervention model by the guidance counselors, intervention coordinator, and/or
teacher recommendation.
Program Implementation
The subjects for the traditional program consisted of 22 sixth-grade participants and 28
Algebra I participants enrolled in the After-School Academy of Success (60 minutes, twice a
week, for 10 weeks). Table 2 shows the participant treatment sample. The After-School
Academy of Success program was developed by the school’s Instructional Leadership team
comprised of the Instructional Administrator, Instructional Coaches, Title I Coach, Title III
Coach (an Access to Core Coach for English Language Learners), Bilingual and Intervention
Coordinator, Content department chairs, and Content Grade level lead teachers. The students
participated in teacher-directed instruction, which included math remedial skills and content
reinforcement, twice a week for 60 minutes. Table 3 shows the intervention schedule. Student
participants were given a pre-test to determine the specific academic standards and skills that
should be addressed during the intervention program. Participants were provided directed-
instruction, group-guided instruction, and peer-interaction activities based on the results from the
assessment. Progress monitoring was conducted periodically to determine participant progress.
Progress monitoring was determined by the individual instructor. In this study, both sixth-grade
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 47

and Algebra I eighth-grade participants completed progress monitoring assessments bi-weekly.
Teachers who participated in the program received 6 hours of professional development by
instructional coaches.
The hybrid group consisted of 28 sixth-grade and 92 Algebra I participants enrolled in the
Math Tutorial Lab 54 minutes, four days per week, and 39 minutes, one day per week for 14
weeks. Table 2 shows the participant treatment sample. Study subjects participated in 54
minutes of computer-assisted instruction three times a week, and 54 minutes one day per week
plus 39 minutes of teacher-directed instruction one day per week. Table 3 shows the intervention
schedule. The Math Tutorial Lab was an intervention program implemented by the school to
enhance the prerequisite skills needed to access grade level math. The computer-based portion
of the intervention was Assessment and Learning in Knowledge Spaces (ALEK), a web-based
assessment and learning system. The program uses adaptive questioning to determine student
level of knowledge and areas of need. Based on assessment, the program instructs students on
topics according to difficulty level and, periodically, reassesses the student as he or she
progresses through the program to ensure that content and skills that have been mastered are
retained. The directed-instruction component of the intervention is composed of explicit
instruction and reinforcement, utilizing teacher- and peer guided activities. Teachers who
participated in the Math Tutorial Lab participated in 12 hours of professional development
provided by district mathematic specialists and school-site instructional coaches.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 48

Table 2
Participant Treatment Sample Itemization
Mathematics
Level

Total Sample
Traditional
Intervention

Hybrid
Intervention
Sixth Grade Course
1

50 22 28
Algebra I 116 24 92

Table 3
Intervention Instructional Schedule
Intervention
Model
Monday Tuesday Wednesday Thursday Friday
Traditional
Intervention
60 minutes
Directed
Instruction
60 minutes
Directed
Instruction

Hybrid
Intervention
54 minutes
Directed
Instruction
39 minutes
Computer
Assisted
Instruction
54 minutes
Directed
Instruction
54 minutes
Computer
Assisted
Instruction
54 minutes
Computer
Assisted
Instruction

Data Collection and Instrumentation
Quantitative instruments used for this study were the CST mathematics scores from the
2010-2011 and the 2011-2012 school years and the Title I expenditure report. The CST
mathematics scores were used to determine the achievement levels and gains of participants’ pre-
and post-intervention as addressed in research questions 1 and 2. Students could have scored
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 49

within five performance bands: Far Below Basic, Below Basic, Basic, Proficient, and Advanced.
The CST was given in May of each school year. The CST assessed student proficiency in
mathematics based on the California Content standards in mathematics. Students performing
below proficient were considered not meeting grade level academic standards.
The Title I Budget Expenditure report provided the cost of the program based on the
categorical funds designated by Title I. The funds were divided into categories that included
teacher pay hours, professional development hours, instructional materials, and general supplies.
The total funds from each category were used for intervention.
Data Analysis
Research Question 1
Is there a significant difference in the performance level of students enrolled in the
traditional intervention programs and hybrid intervention programs?
The results from the 2010-2011 CST scores in mathematics were used as a pre-test
measure to determine the proficiency level of students prior to enrollment in the intervention
programs. The 2010-2011 scores were used to determine student placement in the intervention
program by counselors and instructional coaches.
The researcher performed descriptive statistical analyses using the SPSS software
(Statistical Package for the Social Sciences). An independent Sample t-Test was used to
determine if there was a significant mean difference between the participants’ 2011 Mathematics
CST scores prior to enrollment in the intervention program.
Research Question 2
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 50

Using prior grade CST scores as a control variable for previous math achievement, is
there a statistically significant difference in mean scores in the Mathematics CST scores
between students in the traditional intervention and hybrid intervention?
Students were first administered the mathematics CST in May of the 2010-2011 school
year. These scores were used as a pre-test measure. Finally, students were administered a
second mathematics CST in May of the 2011-2012 school year. The results from the 2011-2012
school year were used as a post-test measure. The assessments were used to determine the
performance level of the students in mathematics and for the comparison of the intervention
programs.
An ANCOVA test was performed to determine if there was an initial and final significant
mean difference and the effect size in the 2012 CST Mathematics scores between students
enrolled in the traditional intervention program and students enrolled in the hybrid intervention
program after completing the intervention.
Research Question 3
Based on a cost-effectiveness analysis, which program will be most cost-effective
between the traditional intervention and hybrid intervention programs as reflected in the
per-pupil cost to increase one point on the CST for Mathematics?
A cost-effectiveness analysis was performed to determine which program had the highest
student gain for the lowest cost. The cost-effectiveness analysis was performed by calculating a
cost-effectiveness ratio. The cost-effectiveness ratio determined the per-pupil cost to increase
one point on the CST for Mathematics. The intervention programs were funded with Title I
funds and the total-cost estimation data was obtained from the Title Budget Expenditures spread
sheet. A total-cost estimation for each program was completed. The total-cost estimation
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 51

included resources (software, textbooks, workbooks, assessment materials, site licenses, and
supplies), salaries, and professional development. The total-cost estimation of each program was
divided by the number of students in the programs to determine a per-pupil cost. The mean
difference between the 2010-2011 CST scores and the 2011-2012 CST scores were used. The
mean difference of each program divided by the average cost per pupil provided the cost-
effectiveness ratio for each program.
Summary
The population for this study was from an urban middle school, grades 6 to 8, with
approximately 1850 students. The school is a Title I school and has a significant percentage of
students identified as low-socioeconomic status. The participants selected for the intervention
program were sixth-grade math and Algebra I students. The participants were selected for the
program if they scored Basic or lower on their 2010-2011 Mathematics CST. For this study,
CST data was collected for the participants enrolled in the intervention programs. In addition,
the total cost estimation for each program was also collected. A t-test was performed to
determine the mean significant difference and effect size of the previous CST scores between the
participants in each program. The t-test was used to decide if there was significant variation in
performance levels between the participants in each program prior to intervention. An
ANCOVA was performed to determine the mean significant difference and effect size of the
CST scores after completing the intervention program. The ANOVCA was used to decide which
program had the highest effect on increasing student performance levels. Finally, a cost-
effectiveness analysis was completed using the mean difference in CST scores within each
intervention and the total-cost estimation of each intervention to determine which program had
the highest cost effectiveness.
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 52

CHAPTER FOUR: ANALYSIS OF DATA
Findings
The purpose of this study was to determine the cost effectiveness between a traditional
mathematics intervention program and a hybrid intervention program by examining the
difference in mean scores on the 2012 CST Mathematics assessment of program participants.
The CST assessed student proficiency in mathematics based on the California Content standards
in mathematics. Students performing below proficient were not meeting grade-level academic
standards. The study analyzed the 2011 pretest scores and the 2012 posttest scores of sixth-grade
and Algebra I students enrolled in the traditional or hybrid intervention programs to determine
the significance of the difference between the intervention programs. In addition, the per-pupil
cost for each program was determined from the Title I Budget Expenditure Report that provided
the categorical funds (designated by Title I) needed to fund the programs.
Organization of Data Analysis
The independent variable in this study was the intervention program models. There were
two variations of the independent variable that included the treatment group that received the
tradition intervention and the treatment group that received the hybrid intervention. The
dependent variable was the 2012 CST mathematics scores. The research questions in this study
focused on the mean difference of the participants between the programs prior to starting the
intervention programs, the mean difference and effect size between the participants after
completing the intervention programs, and the cost effectiveness of the intervention programs.
An initial analysis was conducted using the 2011 CST Mathematics scores to address research
question one. An independent Sample t-Test was conducted to determine if there was a
significant difference between the CST scores of the participants in the traditional intervention
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 53

program and the participants in the hybrid intervention program prior to participating in the
intervention programs.
A one-way ANCOVA (Brace, Kemp, & Sneglar, 2009) statistical analysis was conducted
using the CST mathematics 2011 scores as the pre-test and the 2012 CST mathematics scores as
the post-test to address research question two. The CST mathematics 2011 scores were used as
the pre-test scores and treated as the covariate in the test to control for additional variables that
may influence the results. A between-groups ANCOVA was conducted to determine if there
was a significant difference in means and effect size in CST math scores between the participants
who completed the traditional intervention program and those who completed the hybrid
intervention program.
A final cost-effectiveness analysis was conducted to identify which program had the
highest student gain at the lowest cost to address research question three. The student gain was
determined by the mean difference in the 2011 CST scores and 2012 CST Scores. The total cost
estimation for each program was analyzed using the Title I Budget Expenditure Report and an
average per-pupil cost was estimated by computing the total cost of each program by the number
of participants in each program. A cost-effectiveness ratio was determined for each program by
analyzing the mean difference per-pupil cost.
Research Questions and Associated Hypotheses
In this study, a quantitative approach was utilized. Data were gathered from the
California Standardized Mathematics Test and the Title I Budget Expenditure Report. A
descriptive statistical analysis was performed using the SPSS software (Statistical Package for
the Social Sciences).

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 54

Following are the research questions used to guide this Quasi-experimental study and the
hypothesis for each question:
Research Question 1
Is there a significant difference in the pre-test performance level of students enrolled in
the traditional intervention program and hybrid intervention program?
Hypothesis 1
The data should not reflect a significant difference in prior-year performance levels
between the traditional and intervention program. Sixth-grade students were automatically
enrolled in the program for one semester. For the purpose of this study, only sixth-grade
students with CST achievement levels of Basic or lower were used as participants to match
achievement levels for this study. The data should not reflect a significant difference in prior-
year performance levels between the traditional and intervention programs in Algebra I because
the selection and enrollment process was parallel. Prior year CST data were used for initial
recommendations. Students with CST Mathematics achievement levels of Basic or lower were
enrolled in the programs.
Research Question 2
Using prior grade CST scores as a control variable for previous math achievement, is
there a statistically significant difference in mean scores on the 2012 CST Mathematics
scores between students in the traditional intervention and hybrid intervention?
Hypothesis 2
The CST data will demonstrate a significant mean difference in scores between the
traditional and hybrid intervention model, with the hybrid model showing a higher mean score.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 55

Research Question 3
Based on a cost-effectiveness analysis, which program will be most cost-effective
between the traditional intervention and hybrid intervention programs as reflected in the
per-pupil cost to increase one point on the CST for Mathematics?
Hypothesis 3
There is limited research on the cost-effectiveness of supplementary educational
programs, because of the novelty of computer-assisted programs. Therefore, a reasonable
hypothesis could not be proposed. The observed data will assist in developing a post-study
hypothesis. This hypothesis will be developed based on the student achievement gain on the
CST and the per-pupil cost for each intervention program.
Analysis of Data
This section will present the data analysis for each research question.
Research Question 1
Is there a significant difference in the pre-test performance level of students enrolled in
the traditional intervention programs and hybrid intervention programs?
An independent Sample t-Test at a 95% confidence level was conducted to determine if
there was a significant difference in the mean CST mathematics scores between the intervention
models. There were two intervention models used for this study. The first was a traditional
program that included teacher-directed instruction for 60 minutes twice per week. The second
intervention model included teacher-directed instruction for 93 minutes (54 minutes once per
week and 39 minutes once per week) per week and computer assisted instruction for 54 minutes
three times per week.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 56

Hypothesis 1
There is no significant difference between the 2011 average CST mathematics scores
between the students enrolled in the traditional intervention program and the hybrid intervention
program. The independent variable is the intervention models that included the hybrid and
traditional models. The dependent variable was the 2012 Mathematics CST scores for the
participants in the intervention programs.
Figure 1 presents a graph of the mean CST mathematics scores for sixth-grade level by
intervention model. Figure 2 present a graph of the mean CST mathematics scores for Algebra I
by intervention model.

Figure 1: Sixth-Grade 2011 Average CST Scores by Intervention

314.318
321.107
0
50
100
150
200
250
300
350
6th Grade Traditional 6th Grade Hybrid
Mathematics CST Score
Intervention Model
2011 6th Grade Average CST Scores
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 57

Figure 2: Algebra I 2011 Average CST Scores by Intervention

The data shows there was a 6.789 point mean difference between the sixth-grade
traditional intervention model and hybrid model with the hybrid model showing the highest
average. The data shows that there was an 18.636 point average difference between the Algebra
I traditional intervention model and hybrid model with the traditional model showing the highest
average. Tables 4 and 5 provide the descriptive statistics for the 2011 CST scores by grade level
and intervention model.

Table 4
Descriptive Statistics for the Sixth-Grade 2011 Pretest Mathematics CST Scores by Intervention
Intervention Groups N Mean Standard Deviation
Hybrid 28 321.11 24.963

Traditional 22 314.32 29.404
304.125
285.489
275
280
285
290
295
300
305
310
Algebra Traditional Algebra Hybrid
Mathematics CST Scores
Intervention Model
2011 Algebra Average CST Scores
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 58

The data from the descriptive statistics demonstrated the sixth-grade hybrid intervention
group had a mean CST score of 321.11 and sixth-grade traditional intervention group had a mean
CST score of 314.32. The data from the descriptive statics demonstrated that each score was
different from the mean by an average of 24.963 points in the sixth-grade hybrid intervention
model. In the traditional model, each score was different from the mean by an average of 29.404
points. These data suggested that the hybrid intervention model has a higher mean Mathematics
CST score and less variance from the mean score. The data in Tables 6 and 7 presents the results
for the Independent Sample t-Test.

Table 5
Descriptive Statistics for the Algebra I 2011 Pretest Mathematics CST Scores by Intervention
Intervention
Groups
N Mean Standard Deviation
Hybrid 92 285.489 32.6865

Traditional 24 304.125 24.2887

An independent Sample t-Test was conducted to compare the sixth-grade 2011
mathematics CST scores for participants in the traditional model intervention and the hybrid
model intervention. Levene’s test of homogeneity of variance was checked for the assumption
of homogeneity of variance. The Levene’s test of homogeneity of variances (Table 6) produced
a significance level of p=.870, which is higher than the critical level of 0.05. The results
indicated that the homogeneity of variance was not violated; therefore the independent Sample t-
Test for equal variance assumed was used. Participants enrolled in the hybrid intervention
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 59

(M=321.11, SD = 24.963) model and the traditional intervention model (M=314.32, SD =
29.404) did not differ significantly on the mean CST mathematics scores, t (48) = .883, p = .382
or non-significant (Table 6). The difference between the groups is caused by chance.

Table 6
Independent Sample t-Test for Sixth-Grade 2011 Mathematics
Levene’s Test for
Equality of Variance t-test for Equality of Means
F Sig. t df
Sig.
(2-tailed)
Mean
Difference
Sixth
2011
Equal
variance
assumed

.027 .870 .883 48.000 .382 6.7890
Equal
variances not
assumed
.865 41.234 .392 6.7890

An independent Sample t-Test was conducted to compare the Algebra I 2011
Mathematics pretest CST scores for participants in the traditional intervention and hybrid
intervention. Levene’s test of homogeneity of variance was checked for the assumption of
homogeneity of variance. The Levene’s test of homogeneity of variances (Table 7) produced a
significance level of p=.62, which is higher than the critical level of 0.05. The results indicated
that the homogeneity of variance was not violated; therefore the independent Sample t-Test for
equal variances assumed was used. Participants enrolled in the traditional intervention
(M=304.125, SD = 24.2887) had significantly higher mean CST mathematics scores than the
participants in the hybrid intervention (M=285.489, SD = 32.6865) model, t(114) = .-2.608,
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 60

p = .010 or non-significant (Table 7). The data suggested that the difference between the groups
were not caused by chance and may be caused by the participant selection process.

Table 7
Independent Sample t-Test for Algebra I 2011 Mathematics
Levene’s Test for
Equality of Variance t-test for Equality of Means
F Sig. t df
Sig.
(2-tailed)
Mean
Difference
Algebra I
2011
Equal
variance
assumed

3.555 0.62 -2.608 114.000 .010 -18.6359
Equal
variances
not
assumed
-3.098 47.203 .003 -18.6359

Research Question 2
Using prior grade CST scores as a control variable for previous math achievement, is
there a statistically significant difference in mean scores on the 2012 CST for
Mathematics scores between students in the traditional intervention and hybrid
intervention?
A one-way between-groups analysis of covariance (ANCOVA) test was performed to
determine if there was a significant mean difference between students that participated in the
traditional model intervention program and students that participated in the hybrid model
intervention program for sixth-grade and Algebra I mathematics post-test scores. The 2011 CST
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 61

mathematics scores served as the pre-test and covariate and the 2011 CST mathematics scores
served as the post test.
Hypothesis 2
The CST data will demonstrate a significant mean difference in scores between the
traditional and hybrid intervention model.
Based on Table 8, the Algebra I traditional intervention model has a higher mean CST
score of 301.750 in comparison to the hybrid intervention model with a mean CST score of
290.43. The sixth-grade hybrid intervention model has a higher mean CST score of 319.607 in
comparison to the sixth-grade traditional intervention model of 289.409.

Table 8
Descriptive Statistics for the 2012 Mathematics CST Scores by Intervention and Grade
N Mean Std. Deviation
ALEK (8) 2012 92 290.430 36.7120
TRAD (8) 2012 24 301.750 46.8348
ALEK (6) 2012 28 319.607 32.0640
TRAD (6) 2012 22 289.409 33.2117

According to Brace et al. (2009), an ANCOVA is used when there is a pretest-posttest
design. In this study, the 2011 CST mathematics scores were used as the pre-test and the 2012
CST mathematics scores were used as the posttest; after completing the intervention programs,
the 2011 CST mathematics scores will also serve as the covariate for the analysis. The
ANCOVA allowed the initial variance between the participants in the traditional intervention
model and hybrid intervention model to be equalized so only the effects of the intervention
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 62

programs are measured (Brace et al., 2009). Several criteria for using an ANCOVA were
examined using the sixth-grade mathematics CST scores. The first criterion checked was to
identify if there was a linear relationship between the covariate and the dependent variable
(Brace et al., 2009). A linear relationship was tested by a scatterplot (see Figure 3). Linear fit
lines by subgroup were examined. The scatterplot demonstrates a linear line. The results
suggest that the general distribution of scores indicate a linear relationship between the covariate,
which is the 2011 CST mathematics scores and the dependent variable the 2012 CST
mathematics scores.
The second criterion tested was to determine homogeneity of regression. The
homogeneity of regression assumes that the relationship between the dependent variable and the
covariate is similar for all sub groups (Brace et al., 2009). In this study, the homogeneity of
regression was to test if the relationship between the sixth-grade 2011 CST Mathematics scores
and 2012 CST scores are similar for both the traditional and hybrid intervention. A scatterplot as
shown in Figure 3 was examined to test the homogeneity of regression of slopes. The lines
between the dependent variable and the covariate were parallel and indicated there was no
violation of homogeneity. To verify that the homogeneity of regression of slopes was not
violated, an analysis of between-subjects test was performed (see Table 9).

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 63

Figure 3: Scatterplot of 2011 and 2012 Sixth-Grade CST Mathematics Scores by Intervention

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 64

Table 9
Test of Homogeneity of Slopes for Sixth-Grade Intervention Groups
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Partial Eta
Squared
Corrected Model 24566.655
a
3 8188.885 10.021 .000 .395
Intercept 4423.445 1 4423.445 5.413 .024 .105
6
th
Grade 834.042 1 834.042 1.021 .318 .022
@2011 CST Scores 12675.343 1 12675.343 15.511 .000 .252
6
th
Grade
@2011 CST Scores 454.332 1 454.332 .556 .460 .012
Error 37590.225 46 817.179

Total 4753754.000 50

Corrected Total 62156.880 49

Note: a R Squared = .395 (Adjusted R Squared = .356)

The analysis of between-subjects test produced a significant level of p=0.460 which
verifies that the homogeneity of regression of slopes was not violated. Since the criteria for an
ANCOVA was not violated, a between-subjects ANCOVA was conducted to test the hypothesis
that there will be a significant mean difference in scores between the traditional and hybrid
intervention model. Levene’s test for Equality of Error Variances indicated equal variances (F =
0.14, p = 0.907), which indicates that the homogeneity of variance was not violated (see Table
10). This suggests that the variance between the traditional and hybrid intervention populations
are equal.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 65

Table 10

Levene’s Test of Equality of Variances for Sixth-Grade Mathematics 2012 CST Scores

Dependent Variable: 6th 2012
F df1 df2 Sig.
.014 1 48 .907

The results for the ANCOVA for sixth-grade mathematics are presented in Table 11.
After adjusting for pretest scores, a significant effect was found for the sixth-grade 2012 average
CST mathematics scores, F (1, 47) = 10.186, p= 0.003. Participants in the hybrid intervention
(M = 317.795
a
, SD = 5.396) had significantly higher mean CST mathematics scores than
participants in the traditional intervention model (M = 291.716
a
, SD = 6.093). The results
suggest that the participants in the hybrid intervention increased CST mathematics scores when
compared to participants in the traditional intervention program. Based on eta
2
, 17.8% of the
variance in the sixth-grade 2012 mathematics scores can be explained by the hybrid intervention.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 66

Table 11
Analysis of Covariance (ANCOVA) for Sixth-Grade Mathematics
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Partial Eta
Squared
Corrected Model 24112.323
a
2 12056.162 14.894 .000 .388
Intercept 4296.941 1 4296.941 5.308 .026 .101
@ 2011 CST Scores 12877.440 1 12877.440 15.909 .000 .253
6
th
Grade 8245.158 1 8245.158 10.186 .003 .178
Error 38044.557 47 809.459

Total 4753754.000 50

Corrected Total 62156.880 49

Note: a. R Squared = .388 (Adjusted R Squared = .362)

Several criteria when using an ANCOVA were examined using the Algebra I
mathematics’ CST scores. The first criterion checked: Was there a linear relationship between
the covariate and the dependent variable? (Brace et al., 2009). A liner relationship was tested
by a scatterplot (see Figure 4). Liner fit lines by subgroup were examined. The scatterplot
demonstrated a linear line. The results suggested that the general distribution of scores indicated
a linear relationship between the covariate, which is the 2011 CST mathematics scores and the
dependent variable the 2012 CST mathematics scores.
The second criterion tested was to determine homogeneity of regression. The
homogeneity of regression assumes that the relationship between the dependent variable and the
covariate was similar for all subgroups (Brace et al., 2009). A scatterplot (see Figure 4) was
examined to test the homogeneity of regression of slopes. The lines between the dependent
variable and the covariate intersected in the observed data. The scatterplot indicated a possible
violation of the homogeneity of regression slopes. To verify that the homogeneity of regression
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 67

slopes assumption was violated, an analysis of between-subjects test was performed (see Table
12).

Figure 4: Scatterplot of 2011 and 2012 Algebra I CST Mathematics Scores by Intervention

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 68

Table 12
Test of Homogeneity of Slopes for Algebra I Intervention Groups
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Partial Eta
Squared
Corrected Model 30535.610
a
5 6107.122 4.633 .001 .174
Intercept 4608.460 1 4608.460 3.496 .064 .031
Algebra I 4706.168 2 2353.084 1.785 .173 .031
@ 2011 CST Scores 419.038 1 419.038 .318 .574 .003
Algebra I
@ 2011 CST Scores 4900.158 2 2450.079 1.859 .161 .033
Error 144996.562 110 1318.151

Total 10118786.000 116

Corrected Total 175532.172 115

Note: a. R Squared = .174 (Adjusted R Squared = .136);

The analysis of between-subjects test produced a significant level of p=0.160 which
indicates statistically that the homogeneity of regression of slopes was not violated. Because the
criteria for an ANCOVA was not violated, a between-subjects ANCOVA was conducted to test
the hypothesis that there will be a significant mean difference in scores between the traditional
and hybrid intervention model, with the hybrid having higher CST Algebra I scores. Levene’s
test for Equality of Error Variances (Table 13) indicated equal variances (F = 1.906, p = 0.170),
which indicated that the homogeneity of variance was not violated. This suggests that the
variance between the traditional and hybrid intervention populations are equal.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 69

Table 13

Levene’s Test of Equality of Variances for Algebra I Mathematics 2012 CST Scores

Dependent Variable: 8th 2012
F df1 df2 Sig.
1.906 1 114 .170

The results for the ANCOVA for Algebra I mathematics are presented in Table 14. After
adjusting for pre-test scores, there was not a significant effect found for the Algebra I 2012
average CST mathematics scores, F (1, 113) = 0.116, p= 0.734. Participants in the hybrid
intervention (M = 290.430, SD = 36.7120) did not differ significantly in mean CST mathematics
scores than participants in the traditional intervention model (M = 301.750
a
, SD = 46.8348). The
results suggested that the participants in the tradition intervention program had a greater mean
CST mathematics scores when compared to participants in the hybrid intervention program.
Based on eta
2
, the small variance of 1% in Algebra I 2012 mathematics scores was explained by
the traditional and hybrid interventions that suggested the overall difference between the groups
was very small.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 70

Table 14
Analysis of Covariance (ANCOVA) for Algebra I Mathematics
Source
Type III Sum
of Squares df
Mean
Square F Sig.
Partial Eta
Squared
Corrected Model 24847.630
a
2 12423.815 9.317 .000 .142
Intercept 33518.310 1 33518.310 25.136 .000 .182
@8th2011 22410.566 1 22410.566 16.806 .000 .129
Group_A 154.606 1 154.606 .116 .734 .001
Error 150684.542 113 1333.492

Total 10118786.000 116

Corrected Total 175532.172 115

NOTE: a. R Squared = .142 (Adjusted R Squared = .126)

Research Question 3
Based on a cost-effectiveness analysis, which program will be most cost-effective
between the traditional intervention and hybrid intervention programs as reflected in the
per-pupil cost to increase one point on the CST for Mathematics?
A cost-effectiveness analysis was performed to determine which program had the highest
student gain for the lowest cost. The cost-effectiveness analysis was performed in three steps.
The first step completed a total-cost estimation for each program, by using the Title I
Expenditure Report. The second step determined the per-pupil cost of the program, by dividing
the number of participants in the programs by the total cost. The final step determined the cost-
effectiveness ratio by using the mathematics CST mean difference between the pre- and post-
assessment for each intervention program and dividing by the per-pupil cost. The cost-
effectiveness ratio indicated the amount it will cost to increase one point per pupil on the CST
mathematics assessment (Ashdown & Hummel-Rossi, 2002).

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 71

Hypothesis 3
There is limited research on the cost benefits or effectiveness of supplementary
educational programs, because of the novelty of computer-assisted programs. Therefore, a
reasonable hypothesis could not be supposed until results were analyzed. The post-study
hypothesis suggested, overall, the hybrid intervention was most cost effective in increasing
average student achievement by one point for less cost.
A cost-effectiveness analysis was conducted to determine the average per-pupil cost to
increase one scale point on the CST for Mathematics. The Title I Expenditure report was
utilized to determine the total cost for each intervention program (see Table 15). The traditional
intervention occurred outside of the normal work day; therefore, additional funding was
allocated for teacher salaries. The computer assisted program did not include the Teacher Salary
supplemental rate because the intervention occurred during the normal work hours. In addition,
all professional development was provided during professional development time. The average
per-pupil cost (Table 16) was calculated by dividing the total number of students enrolled in the
programs by the total cost of the programs. The total enrollment for the traditional intervention
program included 206 students and 46 were used for the purpose of this study. The hybrid
intervention program had a total enrollment of 337 students and 120 were used for the purpose of
this study. Based on the total cost and total enrollment, the average per-pupil cost for the
Traditional intervention program was $81.52 and the average per-pupil cost for the Hybrid
intervention program was $33.75.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 72

Table 15
Summary of Title I Expenditure Report by Intervention
Commitment Item Name Traditional Intervention Hybrid Intervention
Teacher-Salary
Supplement/Other

$16,421.06

Instructional Support Materials
General

$372.69 $11,375
Total $16,793.74 $11,375

Table 16
Average Per-Pupil Cost by Each Intervention Program
Cost
Traditional
Intervention
Hybrid
Intervention
Total Cost

$16,793.74 $11,375
Total Number of
Students

206 337
Average Per-Pupil
Cost $81.52 $33.75

The cost-effectiveness ratio was calculated by dividing the average per-pupil cost by the
mean difference between the pre-test and post-test from each intervention (see Table 17). The
results from the cost-effectiveness analysis (see Table 18) indicated the traditional program cost
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 73

of $-6.19 per pupil as well as a decrease of one point on the CST for mathematics. The negative
mean difference and negative cost-effectiveness ratio for the traditional program indicated the
traditional program was a deficit of $6.19 per pupil and caused a decrease of one point on the
CST for mathematics. The hybrid intervention program cost $9.81 per pupil to increase one
point on the CST for mathematics.

Table 17
Descriptive Differences by Intervention
Paired Differences

Mean
Std.
Deviation
Std.
Error Mean
Pair 1 Traditional Intervention 2012
Traditional Intervention 2011

-13.1522 35.7634 5.2730
Pair 2 Hybrid Intervention 2012
Hybrid Intervention 2011

3.4417 38.5504 3.5192

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 74

Table 18
Cost-Effectiveness Ratio by Intervention Group
Cost Traditional Intervention Hybrid Intervention
Average Cost per
Pupil

$81.52 $33.75
Mean Scale
Difference

-13.1522 3.4417
Cost-effectiveness
Ratio per average
point gain $-6.19 $9.81

Summary
An independent Sample t-Test was conducted to test Hypothesis 1: There should not be a
significant difference in prior year performance levels between the traditional and hybrid
intervention programs because the selection and enrollment process only included students
scoring Basic or lower on the CST for mathematics. A preliminary check was conducted to
ensure there was no violation of homogeneity of variance and equal variance could be assumed.
The results for sixth grade indicated that there was not a significant difference in mean scale
scores of the 2011 CST for Mathematics between participants enrolled in the traditional
intervention program and hybrid intervention program. At α=.05, t = 0.883, and p= 0.382, the
hypothesis was accepted. The data indicated there was not a significant difference in CST
mathematics scores for sixth-grade participants in the hybrid intervention model and traditional
intervention model prior to completing the intervention programs. The results suggested that the
participant selection process yielded participants with close equivalence in mathematics
achievement with insignificant variance between the groups. The results for Algebra I indicated
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 75

that there was a significant difference in mean scale scores of the 2011 CST for Mathematics
between participants enrolled in the traditional intervention program and hybrid intervention
program. At α = 0.5, t = -2.608, and p= .010, the hypothesis was rejected. The data indicated the
traditional intervention group had statistically larger mean scale scores when compared to the
hybrid intervention group. The results suggested the participant selection process did not yield
participants with close equivalence in mathematics achievement. There was significant variance
in scores between the two groups.
A one-way between groups analysis of covariance (ANCOVA) utilizing the 2011 CST
for Mathematics as a pretest and covariate was conducted to test Hypothesis 2: The CST data
demonstrated a significant mean difference in scores between the traditional and hybrid
intervention model, with the hybrid model showing a higher significant effect size. The
covariate was used to control for previous math achievement. The independent variables were
the intervention models that included the traditional intervention and hybrid intervention. The
dependent variable was the 2012 CST mathematics scale scores.
Preliminary criteria were checked to ensure that there were no violations in assumptions
of linearity, homogeneity of regression of slopes, and homogeneity of variance. The sixth-grade
results were analyzed and after adjusting for mean scale scores the results indicated that there
was a significant difference on the sixth-grade 2012 CST Mathematics mean scale scores
between participants enrolled in the traditional program and hybrid program. At α = .03, F (1,
47) = 10.186, p= 0.003, partial η² = .178, indicated participants completing hybrid intervention
had statistically significant higher mean scale scores than participants completing the traditional
intervention. Based on eta
2
, 17% of the variance in the 2012 sixth-grade CST scores can be
explained by the hybrid intervention. The hypothesis was accepted. The data suggested that
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 76

participants that completed the hybrid intervention had a higher impact on increasing sixth-grade
CST mathematics scores in comparison to the traditional intervention.
The preliminary criteria was checked to ensure no violations of assumptions for the
Algebra I scores indicated a possible violation of homogeneity. The homogeneity of regression
was verified by an analysis of between-subjects test. At p= 0.160, the result indicated the
homogeneity of regression of slopes was not violated. The Algebra I results from the ANCOVA
were analyzed and after adjusting for mean scales scores, the results indicated that there was no
significant difference on the Algebra I 2012 CST Mathematics mean scale scores between
participants enrolled in the traditional program and the hybrid program. At α = .03, F(1,113)=
0.116, p= 0.734, partial η² = .001, indicated that participants completing the hybrid intervention
did not have statistically significant higher mean scale scores than participants completing the
traditional intervention. Descriptive statistics demonstrated participants in the traditional
intervention had higher mean scale scores. Based on eta
2
, the very small variance of 1% in the
2012 Algebra I CST scores can be explained by the traditional intervention. The hypothesis was
rejected. The data suggested the hybrid intervention did not have a significant impact on
increasing Algebra I CST mathematics scores in comparison to the traditional model.
A cost-effectiveness analysis was performed to develop a post study Hypothesis: The
hybrid intervention was most cost effective in increasing average student achievement by one
point for less cost. A post-study Hypothesis was conducted because there was limited research
on the cost-effectiveness of computer-assisted supplementary educational programs. The cost-
effectiveness ratio provided the per-pupil cost to increase student CST mathematics scores by
one-point. The cost-effectiveness analysis was completed in three steps by determining the mean
difference between the 2011 CST mathematics scores and the 2012 CST mathematics scores for
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 77

both intervention groups, the per-pupil cost, and calculating the cost-effectiveness ratio using the
mean difference and per-pupil cost. The cost-effectiveness ratio demonstrated the traditional
intervention program had a deficit cost of $-6.19 per pupil and decreased one point on the CST
for mathematics. Participants in the traditional intervention had a decrease in mean CST
mathematics scores. The hybrid intervention cost $9.81 per pupil to increase one point on the
CST for mathematics. The results from the cost-effectiveness analysis suggested that the hybrid
intervention was more cost effective by $16 per pupil to increase one point on the CST for
mathematics.
The results suggested overall the hybrid intervention model has a higher impact on
increasing students’ mathematics scores on the CST in comparison to the traditional intervention
program. In addition, the hybrid intervention program was most cost effective when compared
to the traditional intervention program. The conclusions and implications of the data analysis
will be presented in Chapter Five.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 78

CHAPTER FIVE: FINDINGS, CONCLUSIONS, AND IMPLICATIONS
The purpose of this study was to compare and evaluate the effectiveness of a hybrid and
traditional intervention program to determine the most cost-effective program that has the
highest student achievement gain based on the per-pupil cost between a traditional intervention
model and hybrid intervention model. The school used in the study incorporated both
intervention programs as methods to increase the proficiency of students scoring basic or lower
on the CST for mathematics. The targeted grade levels for the hybrid intervention were sixth-
grade students and Algebra I students. The traditional intervention program was utilized for
sixth-, seventh-, and Algebra I students. For comparison, only sixth-grade and Algebra I
students were used for this study.
This chapter consists of six sections: (1) Summary of the Study; (2) Findings of the
Study; (3) Conclusions of the Findings; (4) Implications of the Study; (5) Future Research; and
(6) Summary of the Study.
Summary of the Study
No Child left Behind (2002) created a high-stakes testing situation in most public schools
by intensifying focus on school-level accountability. According to NCLB, by 2014, 100% of
students were to be proficient or higher in mathematics. Currently, districts and school are not
meeting the academic targets set by NCLB. There is a significant mathematics achievement gap
between current performance in mathematics and the academic targets. In an effort to improve
student achievement in mathematics, the federal and state governments provided schools with
additional funds to support supplementary education programs to foster student achievement.
Due to the current fiscal climate, schools had the task of adopting research-based effective
intervention programs that were most cost-effective. Ideally, the goals of the intervention
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 79

programs would service as many students as possible for the highest achievement gain with the
lowest per-pupil cost. Many schools adopted computer-assisted programs to service the high
need; however, computer-assisted programs may not be effective for all students. Traditional
intervention models were also considered. Computer-based and hybrid intervention models
demonstrated effectiveness. Therefore, the ultimate goal was to adopt the program that has the
lowest per- pupil cost to increase student proficiency on the Mathematics CST by one-point. The
research questions addressed in this study are presented:
1. Is there a significant difference in the performance level of students enrolled in the
traditional intervention programs and hybrid intervention programs?
2. Using prior grade CST scores as a control variable for previous math achievement, Is
there a statistically significant difference in means scores on the 2012 CST Mathematics
scores between students in the traditional intervention and hybrid intervention?
3. Based on a cost-effectiveness analysis, which program will be most cost-effective
between the traditional intervention and hybrid intervention programs as reflected in the
per-pupil cost to increase one point on the CST for Mathematics?
The literature review established that a cost-effective intervention is needed to improve
student proficiency in mathematics and meet the academic targets set by NCLB (2002). The
purpose of establishing NCLB was to provide equitable educational opportunities to all students
and close the achievement gap for all students. The literature suggested that there is a national
achievement gap for students in the area of mathematics (NAEP, 2011). Districts and schools
have responded to the need by utilizing the Response to Intervention (RTI Action Network, n.d.)
model. Response to Intervention is a system that provides early intervention to students that are
at risk of failure through research-based instruction and progress monitoring (Fuchs & Fuchs,
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 80

2006). Marzano’s Strategies for Effective Instruction (2001 as cited in Hoover, 2006) and
Gasser (2011) Strategies for Critical Thinking are research-based strategies that are often used in
intervention programs. The research supported Curriculum-Based Assessment (CBA) and
Curriculum-Based Measurement (CBM) as effective systemic progression monitoring tools
(VanDerHeyden & Burns, 2005). Current adopted intervention programs that use research-based
strategies and effective progress monitoring are hybrid programs. Hybrid programs use both
computer-assisted learning and traditional classroom instruction (Tuckman, 2001). According to
Clayton et al. (2010), learning environment has an effect on motivation to learn. The hybrid
programs provided individualized support through a variety of instructional strategies, frequent
progress monitoring, and specific feedback. Research demonstrated positive effects of hybrid
intervention programs (Sinclair et al., 2003). There is limited research on the cost-effectiveness
of hybrid supplementary educational programs.
The purpose of this study was to compare and evaluate the effectiveness of a traditional
intervention program and a traditional intervention program in a middle school setting for
increasing math achievement. In addition, the total estimated cost for each program was also
collected using the Title I Expenditure Report. The sample came from a middle school in Los
Angeles County of approximately 1850 students in sixth to eighth grades six. The school is
identified as a Title I school, because 79% of the population is identified as a low socioeconomic
status. The sample consisted of sixth- and eighth-grade students that scored basic or lower on
the 2010-2011 CST for mathematics. For the purpose of this study, 2011 and 2012 mathematics
CST data was collected for the students enrolled in the traditional and hybrid intervention
programs.
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 81

The participants used for the study scored Basic or lower (<350 scale score) on the CST
for mathematics. The participants for traditional intervention were selected by the counselor,
coordinators, and coaches based on their 2011 Mathematics CST scores. There were 50 sixth-
grade participants and 116 eighth-grade participants used for the study. The traditional
intervention included 22 sixth-grade participants and 24 Algebra I participants. The Algebra I
participants were selected for the hybrid intervention program by the counselor, coordinators,
and coaches based on their 2011 Mathematics CST scores. All sixth-grade students are enrolled
in the hybrid intervention once a semester. For purpose of this study only the students that
scored basic or lower on the 2011 Mathematics CST were used. The hybrid intervention
program included 28 sixth-grade participants and 92 Algebra I participants. The traditional
intervention participants participated in 60 minutes of direct instruction twice per week for 10
weeks. The hybrid intervention participants participated in 54 minutes of computer-assisted
instruction three times per week and 54 minutes per week, plus 39 minutes per week of direct
instruction for 14weeks.
Findings of the Study
The study utilized a pre-test and post-test design. The study design was used to
determine a cause and effect relationship between the intervention models and student
proficiency on the CST for mathematics. In addition, the study compared and evaluated the
intervention model that was most effective in increasing student proficiency and the most cost
effective. The hybrid intervention incorporated the ALEK computer-assisted program.
An independent Samples t-Test was conducted to analyze if there was a difference
between the participants in the intervention groups before completing the intervention program.
The results from the independent Samples t-Test suggested that there was no significant
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 82

difference in the sixth-grade 2011 CST scores and there was a significant difference in the
eighth-grade 2011 CST scores.
A between-groups ANCOVA was conducted to analyze the intervention that had the
highest effect and the effect size. Preliminary analysis was conducted to ensure that the
assumptions of linearity, homogeneity of regression of slopes, and homogeneity of variances
were confirmed. Using the 2011 CST Mathematics scores as a covariate, the results of the
ANCOVA indicated that the hybrid intervention of sixth grade had statistically significant
increased scores in comparison to the tradition intervention. The Algebra I hybrid intervention
did not have statistically significant increased scores in comparison to the traditional
intervention. The adjusted means demonstrated that the tradition intervention in Algebra I has a
higher mean CST mathematics score.
A cost-effectiveness analysis was conducted to determine the per-pupil cost for a one
point increase on the CST for mathematics. The cost-effectiveness analysis was conducted by
using descriptive statistics to determine the CST mean difference between the 2011 and 2012
CST scores for mathematics, calculating the per-pupil cost, and calculating the cost-effective
ratio using the mean difference and per-pupil cost. The results indicated that the cost-
effectiveness ratio was most cost-effective for increasing student proficiency in mathematics
when compared to the traditional intervention program.
Conclusions of the Study
The first research question focused on whether there was a significant difference in the
performance level of the participants for the traditional and hybrid intervention programs. The
results from the independent Samples t-Test indicated that there was no significant difference in
the performance level between sixth-grade participants enrolled in the hybrid intervention and
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 83

traditional intervention programs. The mean scale score for sixth-grade participants on the CST
for mathematics enrolled in the traditional intervention program was 314.318, and the mean scale
score the sixth-grade participants enrolled in the hybrid intervention program was 321.107. The
results indicated that there was a significant difference in the performance level between Algebra
I students enrolled in the hybrid intervention and traditional intervention programs. The mean
scale score for Algebra I participants on the CST for mathematics enrolled in the traditional
intervention program was 304.125 and the mean scale score the Algebra I participants enrolled in
the hybrid intervention program was 285.489. The difference identified between the Algebra I
students could be due to sampling error. There is a drastically larger sample of participants
enrolled in the hybrid intervention program (92) in comparison to the participants enrolled in the
traditional intervention program (24). The traditional intervention program occurs after school
and is voluntary for students. The hybrid intervention program is a mandatory elective that
occurs during the school day. All of the students that were recruited for the program did not
enroll. The sample size and the voluntary basis for the traditional intervention program may
have caused a sampling error. The sixth-grade intervention programs have closer sample sizes of
28 and 22 for the hybrid and traditional programs respectively.
The second research question focused on whether there was a statistically significant
difference in mean scale score on the 2012 CST for mathematics between the participants in the
hybrid intervention program and traditional intervention program with the hybrid program
having the higher affect size. Hybrid programs utilize both computer assisted instruction and
traditional instruction (Tuckman, 2001). Hybrid programs have been shown to increase basic
knowledge skills and improve critical thinking skills (Sinclair et. al., 2003). The results from the
ANCOVA for sixth-grade mathematics indicated a statistically significant difference in the mean
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 84

scale scores on the CST mathematics between the hybrid intervention group and traditional
intervention group. The results suggested that the hybrid intervention had a significant impact
on the sixth-grade CST mathematics scores. The mean difference between the traditional and
hybrid intervention sixth-grade 2011 Mathematics CST scores was 6.789, with the hybrid
program having the higher mean score. The mean difference between the traditional and hybrid
intervention sixth-grade 2012 Mathematics CST scores was 30.198 with the hybrid program
having the higher mean score. The results indicated that the participants in the hybrid
intervention improved significantly higher on the 2012 Mathematics CST in comparison to the
participants who completed the traditional intervention. The results suggested that the hybrid
intervention was most effective for increasing student performance on the Mathematics CST
with 17% of the impact due to the hybrid intervention program.
The results from the ANCOVA for Algebra I mathematics did not indicate a significant
difference in the mean scale scores on the CST mathematics between the hybrid intervention
group and traditional intervention group. The results suggested the hybrid intervention did not
have a significant impact on the Algebra I CST. The hybrid intervention did not have
significantly higher mean scores. The traditional intervention had a higher mean score in
comparison to the hybrid intervention; however, the traditional intervention was not significantly
higher in mean CST mathematics scores. The mean difference between the traditional and
hybrid intervention in Algebra I 2011 Mathematics CST scores was 18.636 with the traditional
program having the higher mean score. The mean difference between the traditional and hybrid
intervention Algebra I 2012 Mathematics CST scores was 11.315 with the traditional program
having the higher mean score. The results indicate that the participants in the hybrid intervention
performed higher on the 2012 CST Mathematics in comparison to the 2011 CST Mathematics.
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 85

The results suggested that the hybrid intervention was effective in improving student
performance on the Mathematics CST from the previous year.
The third research question addressed the cost-effectiveness of each intervention program
to identify which program had the highest student gain for the lowest cost. A cost-effectiveness
ratio was calculated to determine the per-pupil cost to increase one point on the Mathematics
CST. The mean change in CST scores and total cost analysis was used to calculate the cost-
effectiveness ratio. The results from the cost-effectiveness analysis indicated that the hybrid
intervention cost $9.81 per pupil to increase one point and the traditional intervention had a
deficit cost of $-6.19 per pupil. The deficit cost of the traditional intervention program indicated
that participants enrolled in the traditional intervention model decreased proficiency causing a
$6.19 deficit per pupil. The results suggested that the hybrid intervention model was most cost
effective in providing the highest student gain for the lowest cost.
In conclusion, the findings for Research Question 1 suggested that offering intervention
during the school day will ensure more accountability of the students for participation, because
participation will be mandatory and not optional. The findings for Research Question 2
suggested that the hybrid intervention is most effective in increasing student proficiency in
mathematics. The hybrid intervention utilizes both computer-assisted instruction and direct
instruction. Students in the hybrid program were able to benefit from the individualized support
and frequent progress monitoring and feedback of the computer-assisted instruction. In addition,
students were able to benefit from the direct interaction of the traditional program. The hybrid
intervention supported the learning environment preference of most learners. The motivational
beliefs and favored learning strategies influence student choice of learning environment (Clayton
et al, 2010). In addition, the hybrid intervention supported the conceptual knowledge and
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 86

procedural knowledge through diverse learning activities and platforms, which are necessary to
master math concepts (Kilpatrick et al., 2001). The direct instruction was a critical component of
the hybrid intervention. The non-significant effect of the hybrid intervention on math
proficiency for Algebra I could be related to the direct-instruction component. The direct-
instruction component did not have a common program or unified lessons. The findings for
Research Question 3 suggested that the hybrid intervention program was most cost-effective.
The overall findings from the study suggested that the hybrid program will be most cost-effective
to implement in the middle school setting to improve student performance in mathematics.
Implications of the Study
Meeting the challenging academic targets of NCLB (2002) in mathematics and
addressing the achievement gap made the necessity for intervention significant. In addition, a
strained fiscal climate has made the adoption of a cost-effective intervention program priority.
The implications for practice of the implementation of hybrid-intervention mathematics
programs in a middle school setting are based on the research findings and conclusion. This
study suggests three primary implications for practice: (1) develop a standards-based curriculum
that utilizes effective strategies that enhance cognitive knowledge and procedural knowledge in
the direct-instruction component of the hybrid intervention; (2) provide intensive and sustained-
professional development for teachers to ensure the program is implemented with fidelity; and
(3) schools should budget funding for intervention programs that have evidence to support
positive effect.
Effective intervention should not be perceived as punitive but corrective (Ketterlin-Geller
et al., 2008). Elements of effective intervention include the development of foundational
mathematical concepts and skills (Ketterlin-Geller et al., 2008). Effective mathematics
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 87

interventions require the development of conceptual knowledge which allows students to
develop in-depth understanding of the relationships between mathematical concepts (Burns,
2011). An effective intervention also requires the procedural knowledge (the process of solving
the problem) of the mathematical concepts (Burns, 2011). The computer-assisted ALEK
mathematics program provides individualized, targeted learning based on the knowledge level of
the students and sequences instruction accordingly. ALEK provides the essential components of
effective instruction; however, the program does not address the need of direct interaction
needed by some students. Learning environment preference is directly related to motivational
orientation (Clayton et al., 2010). For this reason, the direct-instruction component of the hybrid
intervention model is essential to fostering students understanding and application of the
mathematical concepts. Teachers of the intervention program need quality time for collaboration
to develop a standards-based curriculum that utilizes effective strategies that enhance cognitive
knowledge and procedural knowledge for the direct-instruction component.
A suggested method that will allow teachers to develop effective lessons is the lesson-
study approach. The benefit of lesson study is the process enables teachers to become self-
reflective and self-critical (Hird, Larson, Okubo, & Uchino, 2014). In addition, it provides
teachers with the necessary collaboration time needed to develop effective lessons. The four
steps involved in the process are collaborative planning, lesson observation, analytic reflection,
and ongoing revision (Hird et al., 2014). The process allows teachers to see the practice of other
teachers, use this knowledge to develop their own practice, and provide constructive criticism to
support the instruction of other teachers. The ALEK program could work in conjunction with
the lesson-study protocol. The ALEK program provides teachers with detailed assessment
reports for each student in the class. The reports provide the teachers with proficiency levels of
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 88

each mathematics target. These reports can be incorporated into the collaborative planning
component of the process. This would allow lessons to include the essential components for
mathematics instruction and address the needs of the students. The lesson-study process would
allow teachers to deliver effective instruction and support to the students. The lesson-study
process would ensure that the direct-instruction component of the hybrid intervention model is
complementary to the computer-assisted component. The combination would create a synergetic
effect in increasing student proficiency in mathematics.
The teachers of the hybrid intervention model were provided 12 hours of professional
development on how to use the ALEK program and how to incorporate the suggested lessons
into the direct-instruction component. Providing only the initial 12 hours of professional
development assumes that once is enough time to learn, practice, and effectively develop the
capacity to effectively implement a new concept and skill. Providing effective professional
development to teachers will ensure the program is implemented with fidelity. Effective
professional development should focus on content knowledge, which includes subject-matter
content and pedagogy content; active learning in which teachers are engaged in discourse,
practice, and planning; and coherence to ensure logical and unified activities (Garet, Porter,
Desimone, Birman, & Yoon, 2001). An additional component of effective professional
development is providing ongoing professional development (Garet et al., 2001). Effective
professional development can have a positive impact on instructional practices. Teachers who
implement the hybrid intervention should be provided the intensive- and sustained-professional
development throughout the duration of the program from a facilitator who is competent in the
program, subject matter, and adult learning theory. Professional development opportunities
should be provided to the teachers on a continuous basis.
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 89

Intervention is a response to closing the achievement gap in mathematics. Funding is
provided to schools for implementing supplementary educational programs. In the current fiscal
deficit, budgeting money for specific intervention programs should be unquestionable. Schools,
however, should only budget for intervention programs that have evidence of success. This will
alleviate inefficient expenditures. A criterion of the cost-effectiveness analysis is to only analyze
programs that are successful in achieving the goals of the organization (Hummel-Rossi &
Ashdown, 2002). Districts and school sites should investigate programs and budget for
programs that have significant evidence of increasing student proficiency.
The hybrid intervention program can be highly effective and have a significant impact on
student achievement in mathematics by incorporating lesson study to assist teachers with
effective instruction during the direct-instruction component. In addition, providing intensive-
and sustained-professional development that will support teachers in development of best
practices is necessary. The implementation of lesson study and effective professional
development can significantly enhance the impact of the intervention.
Future Research
Based on the review of the literature and findings from the study, the following
recommendations for future research are suggested:
1. This study examined only one type of hybrid intervention program integrating the
computer-assisted program ALEK. Future studies need to examine the effectiveness of
other hybrid programs. During this study there was limited research on computer-
assisted programs. With emerging computer-assisted programs and the latest research on
learning theories and effective math instruction, additional hybrid intervention programs
should be studied. In addition, possible variables with the hybrid intervention in the
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 90

study may have an effect on the results. Only the computer-assisted component of the
intervention instruction was consistent because of the applications of the program;
however, the instruction in the traditional component of the intervention was inconsistent
among the teachers. This lack of uniformity may affect the results of the intervention.
2. This study was limited to a small sample. Further research needs to use a larger sample
size and multiple grade levels. The study examined one Title I middle school in an urban
school district and a small sample size of 166 sixth-grade and Algebra I participants. The
study did not include seventh-grade students. The sample only accounted for 9.19% of
the total population, 16.6% of the sixth-grade population scoring Basic or lower, and
30.3% of the Algebra I population scoring Basic or lower. In the study, the hybrid
program significantly affected sixth-grade mathematics students, however, not the
Algebra I students. Expanding the research to a larger sample size and multiple grade
levels may alter the results of the programs. Larger sample sizes are more reliable
because they provide a more accurate representation of the population mean (Brace et. al,
2009).
3. This study examined one Title I middle school in an urban school district. The school
site was identified as 79% low socioeconomic status. Extending the research to other
locations and demographic settings may alter the results of the programs. Other
perplexing social and cultural variables may vary between locations and demographic
settings. These variables may affect the motivational orientation of the students. There is
a relationship between learning environments and motivational orientation (Clayton et al.,
2010).
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 91

4. Additional research is needed on the cost effectiveness of intervention programs. There
is limited research on the cost effectiveness of intervention programs that are
implemented in schools. The current fiscal climate makes cost effectiveness a significant
priority. Schools need to insure that money budgeted for intervention programs are
providing a positive investment.
Summary
The purpose of the current study was to compare and evaluate the effectiveness of a
hybrid intervention program and a traditional intervention program in a middle school. This
study also examined the cost-effectiveness of intervention programs. There has been limited
research on hybrid programs and cost effectiveness associated with educational programs.
Participants were selected for the programs based on prior year CST scores of Basic or
lower. The purpose was to select students that were not mastering grade-level standards and
students with equivalent academic proficiency. An independent Sample t-Test was conducted to
determine if there was significant difference in math proficiency levels between the participants
enrolled in the hybrid intervention program and traditional intervention program. The results
from the independent Samples t-Test suggested that there was no significant difference in the
sixth-grade achievement levels and there was a significant difference in the Algebra I
achievement levels. The larger sample size of the hybrid program and voluntary basis of the
traditional intervention may have caused a sampling error.
The effectiveness of the hybrid and traditional interventions were examined using
ANCOVA with the 2011 Mathematics CST scores serving as the covariate. The results for sixth
grade suggested that the hybrid intervention was most effective for increasing student
performance on the Mathematics CST with 17% effectiveness. The results for Algebra I
COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 92

suggested that the hybrid intervention did not have a significant impact on the Algebra I CST.
The participants in the traditional intervention had higher mean scores; however, the hybrid
intervention was effective in improving student performance on the Mathematics CST from the
previous year.
A cost-effectiveness analysis was conducted to identify the program that has the highest
student gain for the lowest cost. The cost-effectiveness analysis determined a cost-effectiveness
ratio for the hybrid and traditional intervention program. The results suggested that the hybrid
intervention model was most cost effective in providing the highest student gain for the lowest
cost. The traditional intervention program decreased proficiency causing a deficit per pupil.
Hybrid mathematics intervention models are more effective in increasing student
proficiency in mathematics compared to traditional intervention model. The hybrid intervention
model provides a dual learning platform for students that are more cost-effective. Implementing
hybrid intervention programs as an academic support elective during the school day can foster
student achievement in mathematics.

COMPARE HYBRID AND TRADITIONAL MATH INTERVENTIONS 93

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Abstract (if available)

Abstract No Child Left Behind (NCLB, 2002), a reauthorization of the Elementary and Secondary Education Act, mandates that districts and schools are accountability for student proficiency in K-12 education. National, state, and local data has demonstrated an achievement gap in mathematics. Institutions are consistently pursuing supplementary educational support programs to support students at risk of not meeting proficiency. Effective intervention is a method to improve student mathematics performance. The deficit fiscal climate has made efforts of implementing effective intervention difficult. Institutions are faced with the task of providing effective intervention with limited financial resources. Recently, institutions have considered hybrid intervention programs that utilize both computer-assisted instruction and traditional instruction. Hybrid intervention models employ a variety of research-based instructional strategies and frequent assessment and feedback to improve mathematics achievement. This study compared and evaluated the effectiveness of a hybrid intervention model and traditional intervention model in a middle school by examining the differences in the mean scale scores of the 2012 CST for Mathematics from a one-way between groups analysis of covariate (ANCOVA) using the 2011 CST Mathematics scores as the control variable. The results from the study suggested that the hybrid intervention had a significant impact on the student performance. Further, a cost-effective analysis was conducted to determine the per-pupil cost to increase student proficiency by one point on the CST for Mathematics. The data suggested that the hybrid intervention is more cost-effective than the traditional intervention.

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Asset Metadata

Creator McKinney, Amber Terry (author)

Core Title The comparison of hybrid intervention and traditional intervention in increasing student achievement in middle school mathematics

School Rossier School of Education

Degree Doctor of Education

Degree Program Education

Publication Date 10/30/2015

Defense Date 03/25/2015

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag hybrid intervention,hybrid mathematics programs,intervention,math,Mathematics,mathematics intervention,middle school,middle school mathematics,OAI-PMH Harvest

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Hentschke, Guilbert (committee chair), Hocevar, Dennis (committee member), Seli, Helena (committee member)

Creator Email amber.mckinney@lausd.net,atmckinnusc@yahoo.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-194891

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