Effective Systems and Instructional Strategies for Mid-Level Mathematics
Achievement

A Doctoral Capstone Project
Submitted to the School of Graduate Studies and Research
Department of Education

In Partial Fulfillment of the
Requirements for the Degree of
Doctor of Education

Brian W. Swartzlander, Jr.
PennWest University
June 2024

MID-LEVEL MATHEMATICS ACHIEVEMENT

© Copyright by
Brian W. Swartzlander, Jr.
All Rights Reserved
June 2024

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Dedication
This work is dedicated to the people who have supported me throughout this long
and challenging journey. It is because of their love, encouragement, and patience that I
have been able to achieve my goals.
To my wife, Amy, who is my constant source of strength and inspiration. Your
selflessness and unwavering support allowed me to complete this project to the best of
my ability. Your love and sacrifice are the foundation of all of my success in this world.
I would be nothing without you.
To my children, Katie and Neil, you are my everything. You have motivated me
throughout this process and every single day since you entered my life. I dedicate this
work to you, and I hope it will serve as a testament to the value of perseverance and the
importance of education.
To my parents, Brian and MaryAnn, I am forever indebted to you for the
opportunities you have provided me and the values you have instilled in me. Your belief
in my abilities and the sacrifices you have made to ensure my successes are
immeasurable. Your encouragement and support have been imperative in shaping the
person I am today.
To my other parents, JD and Betsy, the time and love that you have dedicated to
our family has been invaluable. Your constant support and encouragement were essential
in allowing me to complete this valuable work for our future generations.
This work is a tribute to all of you who have stood by me. I am forever grateful.

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Acknowledgments
With heartfelt gratitude and deep appreciation, I acknowledge:
Dr. Mary Wolf, my faculty capstone committee chair, for her time, consideration,
and devotion to my work. Her encouragement and ability to focus my efforts were
instrumental in the completion of this project.
Dr. Jason Lohr, my local capstone committee member, for his guidance and
encouragement throughout this project. He has mentored me throughout my career, and I
am proud to continue my professional journey under his leadership. I would follow him
anywhere.
Dr. Timothy Scott, the man who taught me the true value of leadership. Thank
you for seeing the leader in me and encouraging me to face my fears.

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Table of Contents
Dedication ....................................................................................................................... iv
Acknowledgments ............................................................................................................. v
Table of Contents.............................................................................................................. vi
List of Tables ................................................................................................................... ix
Abstract ............................................................................................................................ x
CHAPTER I Introduction ................................................................................................... 1
Introduction .................................................................................................................. 1
Background .................................................................................................................. 1
Capstone Focus ............................................................................................................. 2
Research Questions ....................................................................................................... 3
Expected Outcomes ....................................................................................................... 4
Fiscal Implications ........................................................................................................ 4
Summary ...................................................................................................................... 4
CHAPTER II Literature Review .......................................................................................... 6
Literature Review .......................................................................................................... 6
Mathematics Instruction ................................................................................................. 7
Didactical Suitability.................................................................................................. 7
Pedagogical Practice ................................................................................................. 8
Guidance, Structure, and Feedback.............................................................................10
Factors that Lead to Effective Learning .......................................................................12
Reinvention ..............................................................................................................13
Mathematical Systems and Strategies..............................................................................15
Professional Development..........................................................................................16
Instructional Coaching ..............................................................................................18
Professional Learning Communities ............................................................................22
Detracking ...............................................................................................................24
Students with Disabilities ..............................................................................................26
Lower Standards .......................................................................................................27
Misconceptions and Inadequate Training.....................................................................28

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Standards-Aligned Curriculum ...................................................................................29
Role and Importance of the IEP Team .........................................................................30
Economically Disadvantaged Students ............................................................................31
Family Income and Academic Achievement ..................................................................32
Value Assessment ......................................................................................................34
Compulsory School Attendance ..................................................................................35
Mixed Results and the Need for Future Study ..................................................................36
Summary .....................................................................................................................37
CHAPTER III Methodology.............................................................................................. 39
Methodology ...............................................................................................................39
Purpose .......................................................................................................................39
Setting and Participants .................................................................................................41
Research Plan ..............................................................................................................47
Methods of Data Collection ...........................................................................................51
Validity .......................................................................................................................58
Summary .....................................................................................................................60
CHAPTER IV Data Analysis and Results ........................................................................... 62
Data Analysis and Results .............................................................................................62
Data Analysis and Findings ...........................................................................................63
Data Analysis and Findings of the Research Questions .....................................................72
Research Question 1 .................................................................................................72
Research Question 2 .................................................................................................74
Research Question 3 .................................................................................................77
Summary .....................................................................................................................80
CHAPTER V Conclusions and Recommendation ................................................................ 83
Conclusions and Recommendations ................................................................................83
Conclusions .................................................................................................................84
Research Question 1 .................................................................................................84
Research Question 2 .................................................................................................90
Research Question 3 .................................................................................................93
Limitations ..................................................................................................................97
Recommendations for Future Research ...........................................................................99

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Summary ...................................................................................................................101
References .................................................................................................................... 103
APPENDICES............................................................................................................... 111
Appendix A............................................................................................................112
Appendix B ............................................................................................................113
Appendix C ............................................................................................................114
Appendix D............................................................................................................116
Appendix E ............................................................................................................118
Appendix F ............................................................................................................120

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List of Tables
Table 1. Kiski Area School District PSSA Math Proficiency by Grade Level, 2021-

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2022
Table 2. Kiski Area Upper Elementary School Percent Enrollment by Student

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Groups, 2023-2024
Table 3. School A Percent Enrollment by Student Groups, 2023-2024

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Table 4. School B Percent Enrollment by Student Groups, 2023-2024

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Table 5. School C Percent Enrollment by Student Groups, 2023-2024

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Table 6. 2022 PSSA Math Results, Grades 3-8 (Pennsylvania, 2023)

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Table 7. Kiski Area Upper Elementary Math PSSA Results Since 2018

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Table 8. Foundational Categories of School Systems and Methodologies

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Table 9. Curricular Systems and Methodologies

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Table 10. Instruction and Assessment Systems and Methodologies

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Table 11. Professional Learning Systems and Methodologies

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Table 12. Resource Systems and Methodologies

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Table 13. Student Learning Systems and Methodologies

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Table 14. Themes Supporting High-Performance Among All Students

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Table 15. Themes Supporting High-Performance Among Economically

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Disadvantaged Students
Table 16. Themes Supporting High-Performance Among Students with Disabilities

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Abstract
The Kiski Area School District has faced persistent challenges with low achievement on state
standardized mathematics tests, particularly in grades five (5) and six (6). During the 20212022 academic year, 5th-grade math scores exhibited the most significant decline within the
3rd through 8th-grade band, showing a 37% decrease in proficiency from 4th to 5th grade.
This trend was also observed statewide among participants in the Pennsylvania System of
School Assessment (PSSA) exams, which saw a 6.9% decrease in proficiency from 4th to 5th
grade for the 2022 testing period. Despite substantial efforts over the past five years to
improve mid-level mathematics achievement, the district's results have remained minimal.
This study aimed to identify effective strategies employed by high-achieving schools to
enhance mathematics proficiency among 5th graders, especially those with learning
disabilities and from economically disadvantaged backgrounds.
Utilizing quantitative data from the Pennsylvania Department of Education (PDE),
three high-performing schools with at least a 60% proficiency rate on the 5th-grade
mathematics PSSAs and a minimum of 20% economically disadvantaged students were
selected. Qualitative data was then gathered through semi-structured interviews with school
leaders to uncover the methods and processes that contribute to their success. The interviews
revealed three key themes critical to improving student achievement in mid-level
mathematics: the implementation of a guaranteed and viable curriculum aligned with state
standards, a comprehensive assessment system that allows for early identification and
remediation of learning deficiencies, and effective communication strategies to engage
families in supporting their child's academic progress. These findings suggest that a strategic
focus on curriculum alignment, assessment, and family involvement can have a significant
impact on mathematical achievement.

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CHAPTER I
Introduction
The purpose of this qualitative study is to research mathematical systems and
instructional methods that are implemented in schools that have high levels of student
achievement on standardized Mathematics assessments at the 5th grade level. State level
results for PSSA mathematics have consistently shown that the most significant decrease
in proficiency rates among students occurs at the 5th grade level. This data aligns
directly with mathematics proficiency in the Kiski Area School District.
Background
Kiski Area is a rural district located in southwestern Pennsylvania that services
approximately 3,500 students. It is comprised of three (3) K-4 Primary Schools, one (1)
5-6 Upper Elementary School, one (1) 7-8 Intermediate School, and one (1) 9-12 High
School. Although the district has historically performed significantly above the state
average for proficiency on English Language Arts standardized tests, it has failed to
produce similar results in the area of Mathematics. This has resulted in many changes and
initiatives that have had insufficient impact on the achievement of district goals for high
levels of student learning.
In 2022, Pennsylvania state-level results for all students in mathematics showed a
6.9% decrease in proficiency from 4th to 5th grade. At Kiski Area, the negative change
for this testing year was also the greatest between 4th and 5th grade, but to a much larger
degree of 37%. In 2023, state-level results showed a larger decrease in proficiency from
5th to 6th grade at 6.3%, but still a substantial 3.7% decrease from 4th to 5th grade. The
decrease among Kiski Area students from 4th to 5th grade was notably improved to 14%,

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but still represented the largest decrease in mathematics proficiency across grade levels in
the 3-8 student band. State and district data suggests that mid-level mathematics
achievement is sub-par at the 5th and 6th grade level, and failure to meet state standards
for mathematics has a lasting impact on the overall success of the Kiski Area
mathematics department.
Capstone Focus
The Kiski Area Upper Elementary School, has been working with teams of math
teachers to develop curriculum, resources, methods of instruction, common formative and
summative assessments, intervention programs, and enrichment opportunities in an effort
to increase academic achievement at the 5th and 6th grade levels. Although steady
progress has been made over a three-year period, achievement in mid-level mathematics
continues to fall well-below the state average in grade 5, and continues a decreasing trend
in subsequent years. It is reasonable to believe that improvement at this level will
significantly impact the mathematical success of students well beyond the 5th grade
level. This action may initiate change that could have far-reaching possibilities.
The challenging factors at Kiski Area Upper Elementary School include an
increasing special education population, an increasing economically disadvantaged
population, and the transition of students from a self-contained classroom model at the
primary level to an independent departmentalized secondary model at the upper
elementary level. The entity configurations in the Kiski Area School District are unique
when compared to many other districts across the commonwealth. While a K-5 primary
school model accompanied by a 6-8 middle school is the most prevalent structure, a K-4
primary model accompanied by a 5-6 upper elementary and a 7-8 intermediate school are

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employed at Kiski Area. Students from all three primary buildings integrate at the 5th
grade level, and change schools for the first time in their academic careers. These two
factors create a much more significant gap between 4th and 5th grade levels.
Additionally, the large geographic make-up of the district (approximately 102 square
miles) creates substantial barriers for professional collaboration across entities.
A research study that focused on high levels of academic achievement of 5th
grade students on Math PSSAs took place from August 2023 to June 2024. This was a
mixed method study that included quantitative assessment data obtained from the
Pennsylvania Department of Education, specifically focusing on 5th grade mathematical
achievement of special education students and economically disadvantaged students, as
well as qualitative data obtained through formal interviews of school leaders who have
obtained high mathematical performance at the 5th grade level.
Research Questions
This capstone study focused on the following research questions:
Question 1: What instructional strategies and methods do high performing
schools employ to achieve high proficiency rates in mathematics among
all students?
Question 2: What instructional strategies and methods do high performing
schools employ to achieve high proficiency rates in mathematics among
students who are economically disadvantaged?
Question 3: What instructional strategies and methods do high performing
schools employ to achieve high proficiency rates in mathematics among
students with learning disabilities?

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Expected Outcomes
The capstone study provides the research for district review specific to
mathematics curriculum, resources, and philosophy, as well as the evaluation of
mathematical instructional practices that lead to high levels of student learning. The
literature review provides research, strategies, and evidence-based models that the Kiski
Area School District will utilize within the professional learning community process to
support the continuous improvement of student achievement in mathematics.
Fiscal Implications
The financial implications of this capstone project were minimal. Pennsylvania
System of Student Assessment data was obtained through the Pennsylvania Department
of Education website and organized by the researcher. Structured interview questions
were scripted and shared with district Superintendents to obtain formal permission to
conduct research within the district. The interviews will be conducted with identified
building leaders when formal consent is obtained. These in-person interviews should last
approximately 20 minutes. There is a cost associated with taking the time to create,
conduct, and analyze the data obtained through this study.
Summary
A quantitative review of student results on standardized mathematics assessments
led to the examination of schools with historically high achievement in mid-level
mathematics proficiency among economically disadvantaged students, students with
disabilities, and the overall student population. A structured interview with the principals
of these highly-successful schools will assist in obtaining qualitative data specific to the
processes, structures, and instructional strategies and methods that are attributed to the

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mathematical success of students. The overarching goal of this research study is to
obtain relevant data that can be infused in the continuous cycle of improvement process
to increase mathematics proficiency in the Kiski Area School District at the middle
school level.

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CHAPTER II
Literature Review
In 2002, the No Child Left Behind Act was passed by Congress and signed by
President George W. Bush in an effort to bolster the competitiveness of the American
education system on an international scale (No Child Left Behind, 2002). The result was
an increase in the involvement of the federal government in monitoring schools across
the nation for the academic achievement of all students. This law was later replaced with
the Every Student Succeeds Act (ESSA) in 2015. In order to remain compliant with
federal legislation regarding education and remain eligible for federal funding, school
districts across the country were required to evaluate students in both mathematics and
reading using standardized tests in grades three through eight. In the commonwealth of
Pennsylvania, the standardized tests at these levels are called the Pennsylvania System of
School Assessments (PSSAs), and the results of these assessments are reported to the
state and made available to the general public through the Future Ready PA Index.
In addition to ESSA, the STEM Education Act of 2015 was also enacted to
provide more professional development to education professionals related to science,
technology, engineering, and mathematics and address the increasing number of STEMrelated jobs in the United States at the turn of the 21st century. This has not only resulted
in more focus on teaching problem-solving and analytical thinking skills, but it has also
led to increased rigor in mathematical testing provided through standardized testing
services such as the PSSAs. In turn, school administrators and teacher leaders have
concentrated efforts to address K-12 mathematical systems and grade-level instruction to
ensure student growth and high levels of proficiency in essential mathematics skills.

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This chapter is organized into four main sections based on a review of the
literature. First, mathematical instructional practices are analyzed to determine which
pedagogical methods produce high levels of student achievement and in which contexts
these practices are most effective. Second, district approaches and school systems are
explored to determine what strategies currently exist to promote high levels of learning in
the area of mathematics. Next, students with disabilities become the focal subgroup of the
student population, and contemporary methods such as detracking and full inclusion in
the general education curriculum are investigated. Finally, this chapter concludes with an
analysis of the economically disadvantaged subgroup of students and the research on why
this particular group of individuals is statistically more likely not to achieve in the current
educational environment.
Mathematics Instruction
Didactical Suitability
In order to identify effective instruction, one must consider the many factors that
impact teachers, students, and schools in which learning takes place. It is difficult to
identify any one instructional method or practice that is better than another, mainly
because so many teaching practices exist, and those specific techniques vary significantly
from one teacher to the next. The instructional method alone does not have the greatest
impact on student learning, but the combination of instruction and other essential
variables has a much larger influence. The theory of didactical suitability is a holistic
perspective that considers the contribution of multiple variables to optimize instruction.
These factors include teaching, learning, and content, along with the affective
environment and social environment to which students are exposed (Godino et al., 2023,

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p. 14). The impact of instruction cannot be solely attributed to the teacher and the
instructional practice, but rather all of the components of the teaching environment that
formulate each individual child’s perception (Brousseau, 1997).
Learning is the outcome desired by all teachers when instruction is provided;
however, there are many factors for teachers to consider to obtain optimum results.
The didactical suitability of an instructional process is the degree to which such a
process (or a part of it) meets certain characteristics that qualify it as optimal or
adequate to achieve the adaptation between the students’ personal meanings
(learning) and the intended or implemented institutional meanings (teaching),
considering the circumstances and available resources (environment). (Godino et
al., 2023, p. 4)
Particularly in the area of mathematics, the optimization of this process is vital not only
to ensure that a connection is being made between a child’s current reality and the
meaning of the lesson but also to engage learners in the educational process itself, thus
creating a feeling of satisfaction in doing so.
Pedagogical Practice
Instruction can take on many forms. In any scenario, the teacher must consider
multiple factors to determine the approach that will achieve the desired result of student
learning. Dell′Olio and Donk (2007) described instructional choices as “a spectrum, with
lessons controlled by the teacher, such as direct instruction at one end, and increasingly
indirect, open-ended lessons that focus on student exploration, such as the models of
inquiry, at the other end” (p. 439). It is the judgment of the teacher or instructional team
to decide which instructional model or combination of instructional methods is

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appropriate; however, much consideration needs to be devoted to specific factors,
including the content being taught, the research base supporting effective models by
subject (i.e., math computation, reading comprehension, etc.), critical thinking skills
needed, grade-level expectations, and overall student motivation (Dell′Olio & Donk,
2007, p. 440).
Considering the high standards set forth for mathematics in Pennsylvania and the
need for increased proficiency in the mathematics discipline, it has never been more
critical for teachers to implement the best instructional strategies that meet the needs of
their students. Of all variables present in the educational process, teachers have the most
control over their method and approach to instructional practice. Especially in the early
years of mathematical instruction, it is crucial that teachers use the best pedagogical
strategies to ensure high levels of learning and continued student growth. In Hattie and
Zierer’s (2018) research of over eight hundred performance influencers, the collective
efficacy of teachers has the most significant impact on student success (p. 26). Collective
efficacy is the shared belief of teachers in their ability to positively affect students. It is
critical that teachers combine their knowledge and expertise and collectively decide that
they can overcome current obstacles and limitations in the educational setting.
According to Clements et al. (2023), teaching strategies can either be educative or
mis-educative based on their effectiveness in helping students obtain new knowledge and
preparing them for growth in future experiences.
Knowledge of developmental paths in learning trajectories can enhance teachers'
understanding of children's thinking, helping teachers assess children's level of

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understanding and offer instructional activities at the next level and thus offer
meaningful and joyful opportunities to engage in learning. (Clements et al., 2023,
p. 17)
Regardless of the instructional method chosen, it is critical that teachers have a vast
understanding of the content that they teach and the individual needs of their students to
ensure success at the current level of instruction, as well as readiness for the next level of
instruction.
Guidance, Structure, and Feedback
Guidance and structure are two components of the instructional process that vary
significantly from teacher to teacher, lesson to lesson, and student to student. This
variance may be arbitrary or based on an informal assessment of student progress, but the
intentional utilization of these two elements within an instructional lesson can have a
significant impact on results. Horan and Carr (2018) define guidance as:
The interaction between teacher and students, specifically, the amount of
feedback teachers provide in response to students’ questions and learning
difficulties, the quantity and quality of teachers’ responsiveness to students’
questions and concerns, scaffolding provided by the teacher, and how often
teachers ask students questions that are designed to cause students to think more
deeply. (p. 4)
Horan and Carr (2018) go on to define structure as “the purposeful explicitness and
organization of the lesson plan, curriculum, or materials for understanding” (p. 6).
In any given lesson, the degree to which students need guidance and structure will
vary based on multiple factors. Two significant factors are the students’ understanding of

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the new information that is being presented, along with each individual’s past educational
experience and ability to make cognitive connections. Fyfe et al. (2012) specifically
noted that teacher feedback in this process plays a central and invaluable role. “Feedback
during exploratory problem solving prior to instruction facilitates learning for children
with low prior knowledge of a domain. However, children with moderate prior
knowledge benefit more from exploring independently without feedback before receiving
explicit instruction” (p. 1107).
Feedback is any information provided that relates to student comprehension or
performance. Specifically in the area of mathematics, Emily Fyfe and Sarah Brown
conducted meta-analytic research on the effects of feedback on students between the ages
of 6 and 11 years. The research focused on corrective feedback compared to no feedback
when students were presented with math equivalence reasoning in multiple scenarios.
The study concluded that
Feedback had positive effects for low-knowledge learners and negative effects for
high-knowledge learners, and these effects were stronger for procedural outcomes
than conceptual outcomes. Findings highlight the variable influences of feedback
on math equivalence understanding and suggest that models of thinking and
reasoning need to consider learner characteristics, learning outcomes and learning
materials, as well as the dynamic interactions among them. (Fyfe & Brown, 2018,
p. 174)
Every individual continuously uses feedback to make decisions, guide their
actions, and form their own reality. This feedback can be verbal, non-verbal, or
paraverbal. Paraverbal is a term to describe information that is conveyed through the

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tone, pitch, and pacing of verbal communication. From an educational perspective, when
feedback is used, how it is used, and the degree to which it is used has a significant
impact on student learning. Hattie and Timperley (2007) identified three main feedback
questions that can be provided by teachers: “Where am I going? How am I going? Where
to next?” These questions are imperative because the answers “enhance learning when
there is a discrepancy between what is understood and what is aimed to be understood”
(p. 102). Similar to the theory of didactical suitability, in which many variables contribute
to effective instruction, feedback also requires much skill and consideration. The
effectiveness of feedback
Does not merely invoke a stimulus-and-response routine but requires high
proficiency in developing a classroom climate, the ability to deal with the
complexities of multiple judgments, and deep understandings of the subject
matter to be ready to provide feedback about tasks or the relationships between
ideas, willingness to encourage self-regulation, and having exquisite timing to
provide feedback before frustration takes over. (Hattie & Timperley, 2007, p.
103)
Factors that Lead to Effective Learning
Many educators consider the relationships they build with their students to be
critical in producing an internal interest in the subject matter, thus leading to higher levels
of student achievement in that content area. This theory is often compelling to teachers
because it is a major variable that contributes to both the affective environment and social
environment that students experience. When considering the nature of the mathematics

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discipline, data suggests that positive interactions between students and their teachers
may have an adverse effect on the degree to which students enjoy mathematics.
Using the Australian model of Quality of School Life (QSL), which describes
“the quality experienced by students in education places where they feel safe, good, and
motivated,” a study was conducted to determine what specific factors lead to enthusiasm
for learning and satisfaction in mathematics (Aliyev & Tunc, 2015, p. 165). Winheller et
al. (2013) conducted a detailed analysis of 336 elementary students and 272 high school
students in New Zealand to determine if any relationships exists between students’
quality of school life perceptions, students’ attitudes to mathematics, and their effects on
mathematical achievement (p. 49). The study data confirmed that an individual’s interest
in mathematics as a subject and self-efficacy directly contribute to why a student feels
more or less confident with mathematical applications. Additionally,
Factors of ‘satisfaction with’ and ‘enthusiasm for learning’ positively predicted
liking of mathematics, while the perception of a caring teacher and positive peer
interaction all negatively predicted liking of mathematics. Furthermore, the results
showed that liking mathematics itself had negative or zero impact on mathematics
performance. (Winheller et al., 2013, p. 63)
These strong correlations refute the notion that positive student-teacher relationships have
a significant impact on overall student learning.
Reinvention
The traditional approach to teaching mathematics involves the explicit instruction
of a method or algorithm followed by a commonly utilized application for what has been
presented. The application is modeled using examples and the student is then expected to

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replicate the application repetitively until mastery is achieved. In contrast to this deeply
rooted philosophy of math instruction, Lai et al. (2019) explained that a learner-centered
approach has proven more effective because it directly relates to the enthusiasm for
learning factor, as described in Winheller’s study. Lai concluded that “a shift in focus has
led to a move away from teaching number computation as presenting algorithms created
by teachers to engage students in investigative activities such as problem-solving to
construct conceptual understanding for themselves. This shift redirects students learning
from merely memorizing computation procedures…to supporting students to construct
knowledge of the mathematics that underpins the concept and algorithm for themselves”
(Lai et al., 2019, p. 1). She further explained that “the reinvention of mathematics by
students is a basis for deeper understanding of the ‘why’ and also the ‘how’ of
mathematics in solving” (Lai et al., 2019, p. 14).
Summer (2020) came to a similar conclusion when considering mathematical
instruction at an early age. Her research emphasized the importance of skilled teachers
and a focus on student-centered learning. “Up-to-date mathematics teaching requires
teachers to have a profound knowledge of mathematics, didactic knowledge, and an
awareness of possible difficulties” (Summer, 2020, p. 117). Summer further explains
Teachers become active co-designers of future generations through their
activities. They create settings in which pupils can transfer mathematical
understanding across varied contexts and settings. The student-centered
mathematical tasks allow children to develop an understanding of sustainability
issues that enable them to take positive action in their daily lives. (Summer, 2020,
p. 117)

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The development of young mathematical minds through experimentation and
inquiry generates enthusiasm for learning within the discipline. Research indicates that
teaching for relational understanding also enhances skill development. Instead of
spending a lot of time on drills, repeated experiences with various contexts and different
types of activities help with generalization and transfer (Clements et al., 2023). When
teachers focus on meaningful and enjoyable experiences for children, implemented
pedagogical strategies provide the optimal level of guidance and structure, and consider
the affective and social environments of the learning setting, they are better equipped to
interpret what the child is doing, thinking, and constructing, and provide what is
necessary to extend mathematical reasoning.
Mathematical Systems and Strategies
In 2014, the National Council of Teachers of Mathematics (NCTM) identified
five interrelated strands that lead to mathematical proficiency. These strands include
conceptual understanding, procedural fluency, strategic competence, adaptive reasoning,
and productive disposition. When students are able to comprehend and connect math
concepts, utilize meaningful and flexible procedures to solve problems, think logically,
and justify their own thinking, they have “the tendency to see sense in mathematics,
perceive it as both useful and worthwhile, believe that steady effort pays off, and see
themselves as effective learners and doers of mathematics” (National Council of
Teachers of Mathematics, 2014, pp. 7-8).
As discussed in the analysis of instruction that impacts student achievement in
mathematics, there are a large number of factors that contribute to effective systems and
strategies that have the potential to positively impact mathematical proficiency. This

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section will focus on four specific systematic approaches that are commonly
implemented with the intent of producing higher levels of student learning. These
systems include the adherence to quality professional development, incorporation of
instructional coaching, commitment to professional learning communities (PLCs), and
the detracking of students based on perceived learning capacity and ability level.
Professional Development
All public schools in the United States of America require that certified teachers
hold a bachelor’s degree, and many also require the completion of a master’s program
within the first five years of teaching (All Star Directories, Inc., n.d.). In Pennsylvania,
Act 48 of 1999 requires that all Pennsylvania teachers holding a public-school
certification participate in ongoing professional education in the form of 180 continuing
professional education hours in a five-year period (Pennsylvania Department of
Education, 2023a). Additionally, Chapter 49.17 of the Pennsylvania code titled
“Continuing Professional Education,” requires that for strategic planning, a school must
submit a 3-year professional education plan for approval every three years. Before the
school board approves and submits the plan, it must be available for the public to review
and comment on for at least 28 days. The plan should clearly define terms such as
graduate-level courses related to the profession, relevant master's degrees, curriculum
development, and professional conferences (Pennsylvania Department of Education,
2019).
It is clear that the professional development of school staff is significantly valued
when state requirements of districts, schools, and certified teachers are identified. The
collective consensus is that continued learning is essential to providing students with

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high-quality teachers and proven instructional practices; however, there is much to be
learned about what constitutes effective professional development that leads to
sustainable results in the teaching profession.
Garet et al. (2011), based on the results of a two-year study, questioned the
cumulative impact of professional development on teacher knowledge and student
achievement, specifically on the mathematical topic of rational numbers. The study
provided 68 hours of intensive professional development over the course of one year and
an additional 46 hours of continued learning in the second year for each participating
teacher. Students who received instruction from this selected group of teachers minimally
outperformed students who did not receive instruction from trained teachers on a
standardized math exam on the topic of rational numbers. The result of this research
concluded that the professional development process implemented had no statistically
significant impact on teacher knowledge or student achievement (Garet et al., 2011).
Similar to the conclusive results reported in the previous case study, an analysis of
two groups of teachers who participated in a 3-day professional development session over
the summer proved ineffective in changing the current pedagogical methods of
participating teachers. Each session addressed a specific mathematical unit and provided
an additional day of professional development prior to classroom implementation. The
conclusion of the research suggested that teachers may need "long-term professional
development to improve their support for mathematical argumentation practices"
(Roschelle et al., 2010, p. 872). Much like the theory of didactical suitability that was
described in the review of effective instruction, the impact of professional development

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on teacher improvement and student achievement may be dependent on additional factors
and circumstances.
Based on the results of the case studies provided, it is difficult to predict the
success of a professional development plan based on the overall time devoted to the
program. What may be a better predictor of success is the collective ownership, or buyin, of the new strategies and/or philosophies that are being adopted (Garet et al., 2011).
Were teachers and educational professionals involved in the research that led to the
interested professional development? Do the professional development goals align with
the district and school visions, as well as the beliefs and values of those who will be
expected to implement newly attained knowledge? Will time be provided to plan,
implement, and reflect on new methodologies addressed? Learning is the first step, but
school leaders miss the mark to ensure implementation with a high level of fidelity.
Instructional Coaching
The utilization of instructional coaching in the educational process is a relatively
new phenomenon. First introduced in the 1980s by Bruce Joyce and Beverly Showers,
they asserted that professional development does not always transfer to classroom
implementation. Instructional coaching gained traction at the turn of the 21st century with
the passing of No Child Left Behind and the Reading First Initiative (Nugent et al.,
2016).
As described in the previous section, professional development alone has shown
little direct impact on improved teacher knowledge and increased student achievement;
however, there are several studies that suggest instructional coaching may be a viable
supplement to effective professional learning and sustainable implementation. For the

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purpose of this review, we will focus our research on coaching in the mathematics
discipline.
"A mathematics coach is an individual who is well versed in mathematics content
and pedagogy and who works directly with classroom teachers to improve student
learning of mathematics" (Hull et al., 2009, p. 3). Mathematics coaches fulfill a
leadership role and provide assistance to teachers in the areas of professional
development, mathematical content, teaching, and curriculum development; however,
there is little data to suggest that the incorporation of a mathematics coach produces
consistent positive change. The role of the instructional coach is still being examined, but
a study by Campbell and Malkus (2011) demonstrated a correlation between higher
achievement on standardized mathematics tests among students in schools that employed
an elementary mathematics coach. In a period of three (3) years, five (5) school districts
of varying demographics in Virginia identified two (2) to four (4) schools within their
region to be part of the study. A total of 36 schools were involved in the study, and each
school was randomly assigned to have a math coach or be part of the control group with
no math coach. When standardized test scores were analyzed at the end of the 3-year
period, the students enrolled in schools with mathematics coaches had significantly
higher scores than the students in the control group. It was also determined that this
difference was more significant in grades four (4) and five (5) than in grades three (3) and
six (6). There was minimal correlation between math coach implementation and student
achievement in years one (1) and two (2) of the study, suggesting that implementation of
this strategy does not produce rapid results (Campbell & Malkus, 2011).

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In a study by Russell et al. (2020), an instructional coaching model called the
“Tennessee Math Coaching Approach” was analyzed to determine specific coaching
practices that led to positive change and sustainable teacher improvement (p. 442). A
selected sample of 32 individuals were trained in two different methodologies of
instructional coaching, one being the Tennessee approach and the other a more
generalized coaching approach for school reform. The Tennessee approach was a process
of enhancing teacher skills to implement challenging mathematics tasks that encourage
students to think critically about mathematical concepts, while the traditional approach
trained coaches to assist teachers in the substitution of new teaching practices over
traditional methods. Over a period of two years, all participating coaches received
approximately 55 hours of training in their selected coaching model, and a final total of
103 teachers benefited from professional learning with the assistance of the trained
instructional coaches. The conclusion of this research did determine that the Tennessee
method was more impactful in improved instruction, but there were several factors within
this model to which much of the success was attributed. It was determined that “when
coaches had deep and specific conversations with teachers in the context of planning
specific lessons – including attention to content, pedagogy, and student learning –
teachers improved their capacity to maintain the cognitive demand of high-level
mathematics tasks” (Russell et al., 2020, p. 459). In general, collaborative planning with
an instructional coach, along with one-on-one reflection after implementation,
significantly impacted the ability of a coach to build teaching capacity in the area of
mathematics. This study produced specific components of effective instructional
coaching, but there is additional research that highlights other characteristics.

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Hull et al. (2009) identified six characteristics most associated with math
instruction in current classrooms. These include teachers as the primary source of
information, students as passive listeners, rigid instructional structure, the textbook as the
primary curriculum, procedural assessments, and rules and procedures over mathematical
reasoning. In contrast, "the desired traits of envisioned, highly effective mathematics
classrooms" include empowered teachers, an established and implemented curriculum
that is aligned to state standards, the use of multiple instructional strategies, actively
engaged students, and frequent and formative assessment that is inclusive of feedback (p.
17).
Coaching is a process by which an experienced individual supports the
development of learners to improve performance and reach professional goals. In the
education profession, instructional coaching has shown promise in assisting teachers in
their growth and progress. According to Knight (2022),
Coaching is essential for the kind of growth we need to see in
schools…workshops, books, and webinars can provide us with an overview of
ideas, but we only adopt and internalize these ideas when we apply them to our
professional practice…coaches help with each aspect of this kind of learning by
partnering with teachers to establish a clear picture of reality, set emotionally
compelling, student-focused goals, and learn, adapt, and integrate teaching
practices that help teachers and students hit goals. (p. x)
With the proper balance of professional learning and instructional coaching, school
improvement and student achievement are attainable.

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Professional Learning Communities
A professional learning community (PLC) is not a program but rather a
continuous, never-ending process of conducting schooling that has a profound
impact on the structure and culture of the school and the assumptions and
practices of the professionals within it...an ongoing process in which educators
work collaboratively in recurring cycles of collective inquiry and action research
to achieve better results for the students they serve. (DuFour et al., 2010, pp. 1011)
Professional learning communities involve regular, intentional collaboration of teachers
and school leaders in the development of a guaranteed and viable curriculum, the
strategic planning of instructional practices, administration and evaluation of common
formative assessments, and the use of intervention and/or extension to address students
based on proficiency. This is a repeating process that begins with the identification of
agreed-upon essential skills by content or subject area and concludes with the extension
of learning beyond what is expected or the reteaching of skills to ensure student
proficiency (DuFour et al., 2010).
Robert Marzano, an advocate and researcher of professional learning communities
at work, identified critical school- and teacher-level factors that impact student
achievement. School-level factors include a guaranteed and viable curriculum,
challenging goals and effective feedback, parent and community involvement, a safe and
orderly environment, and collegiality and professionalism. Teacher-level factors include
instruction, classroom management, and classroom curriculum design (Marzano, 2003).
Marzano also identified student-level factors that impact achievement; however, many of

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the factors, if not all, are beyond the control and influence of the schools and teachers that
serve them.
A viable standardized curriculum is one that can be implemented in the time
available for instruction. It is virtually impossible for educators to address all state gradelevel standards in a single year, so teachers must make a clear distinction between what is
essential content and what is supplemental content. Once the viable curriculum is
established, teachers must be required to address the identified essential skills so that it is
guaranteed for all students (Marzano, 2002).
Similar to Marzano’s research in the identification of essential skills in the
standardized curriculum, McTighe and Wiggins (2013) describe the importance of
essential questions framed to help students come to an understanding of key ideas and
processes associated with a concept or topic. This is in contrast to the most common
types of questions used in the classroom that merely identify presented information or
check for understanding. Although these non-essential questions serve a purpose and are
often necessary for formative purposes, they often fail to stimulate further thinking,
create discussion or debate, and/or require justification or support (McTighe & Wiggins,
2013).
Professional collaboration is invaluable to the advancement of student learning. It
is through this process that curriculum is developed and implemented, essential skills and
essential questions are identified and targeted, instructional strategies are created and
executed with fidelity, formative assessments are administered and evaluated, student
proficiency levels are determined, and targeted interventions can be applied to ensure that

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all students learn. DuFour et al. (2010) identified four critical questions to guide this
collaborative process:
1. What do we want students to know and be able to do?
2. How will we know if they have learned?
3. How will we address students who have not learned?
4. How will we extend the learning of those students who already know it?
The idea of professional learning communities first came about in the 1960s, and
many business organizations across the globe currently use this collaborative model to
improve productivity, efficiency, and overall customer satisfaction; however, from the
educational standpoint by which the concept emerged, PLCs exist in pockets, and full
commitment to this collaborative process is sporadic (Solution Tree, Inc., 2023). This is
likely due to the substantial commitment of providing the time and creating a school
schedule that allows for a daily focus on the continuous cycle of improvement. Many
educational leaders agree that professional teamwork leads to better teaching and
learning, but they are unwilling to complete the necessary research and fully implement
the strategies required to achieve the results that PLCs offer. Those leaders who have
pledged themselves to follow the PLC process with fidelity commit to the development
of a new educational culture that focuses on what students are learning rather than what is
being taught (DuFour et al., 2010).
Detracking
Detracking can be defined as the placement of students with mixed abilities and
academic achievement in the same classes, with the intention of exposing all students to a
high-quality curriculum (Culver City High School, n.d.). Teachers and school leaders

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have minimal impact on the lives of students outside of the school. They can, however,
do much within the school to ensure a level playing field by providing equal access to
learning opportunities. Detracking, when implemented with fidelity, can significantly
close the achievement gap between those who have access to resources and those who do
not (Burris, 2010). Heterogeneous grouping also allows all students of varying ability
levels the same access to a guaranteed and viable curriculum and high-quality teachers
regardless of their social or economic status (Oakes, 1995).
"More and more educators are recognizing that low-track classes offer a watereddown set of educational opportunities and that denial of educational opportunity is an
unacceptable abandonment of core American values" (Welner & Burris, 2006, p. 98).
Years of tracking research supports that low-level, non-progressive courses should be
phased out. Classes like general math and business English fail to adequately prepare
students for postsecondary opportunities and are less effective than standard courses
(Gamoran, 2009). "Low-track classes depress student achievement, causing students to
fall further and further behind" (Welner & Burris, 2006, p. 93).
In order for detracking to be successful, academic support needs to be considered
and provided for struggling learners in heterogeneous classes (Welner & Burris, 2006).
The common practices necessary for successful mixed-ability settings are differentiated
instruction, teacher response to struggling learners (intervention), and the use of
resources to supplement instruction and adjust to the needs of individual students
(Gamoran, 2009).
Boaler (2006) coined the phrase "relational equity" in his 2006 case study at
Railside School, which is an urban high school in California. The 4-year longitudinal

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26

study produced data showing that students not only increased achievement in
mathematics through the use of mixed-ability grouping but also learned to appreciate
students from different cultures, social classes, genders, and ability levels. The latter was
achieved not through the study of past history and pertinent examples but rather through
exposure to different insights, methods, and perspectives resulting from the collective
problem-solving process (Boaler, 2006, p. 41).
A similar study was conducted on a much smaller scale at a diverse middle school
in South Carolina. The Spartan Middle School services approximately 790 students in
grades six through eight, and the sixth-grade math team at this entity piloted a
differentiated instructional approach throughout the course of one school year. The class
was made up of 18 total students of varying abilities and followed the model of full
inclusion as outlined in the Individuals with Disabilities Act (IDEA). In the conclusion of
this mixed research, it was determined that the differentiated instruction provided
throughout the course resulted in increased student learning as well as positive changes in
students’ attitudes toward the mathematics discipline. When final exams were
administered at the conclusion of the course, 81% of the students showed substantial
growth. Additionally, the results of a student survey showed that 87% of the students in
the class preferred the new class structure when comparing it to the traditional classes
they experienced in the past (Patterson et al., 2009).
Students with Disabilities
Similar to the research on the detracking of students, studies on the effect of
curriculum on students with disabilities significantly show that the lack of exposure to
high standards of learning and lower expectations of student performance result in lower

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27

levels of student achievement (Blank and Smithson, 2014). Although there is much to
consider for students with identified learning deficiencies, their educational experiences
have a significant impact on their progress, as well as their access to future opportunities.
Lower Standards
Math instruction for special needs students tends to take on a different focus from
the education provided to general education students. Many students are losing out on a
meaningful mathematical education because they qualify for special education services.
Providing a rudimentary, watered-down curriculum for students with learning disabilities,
absent of critical thinking skills and higher-order thinking, is not the best solution for the
achievement of basic mathematical skills (Ballin et al., 2022). Instead, educators need to
consider resources and professional learning that enable them to instruct a wide range of
student learning needs.
"Empowerment Math" is a phrase used by Ballin et al. (2022) to describe access
to higher-level thinking. This is the opposite of math instruction as a routine of providing
examples, completing problems, reviewing material, and moving on to the next topic.
The suggestion is to incorporate ten specific principles into daily math instruction that
encourage students to think critically and become intrinsically motivated within the
mathematics discipline. These principles are referred to as the "Nine + One Principles:"
1. Find multiple ways of teaching one concept.
2. Use conceptual mathematics instruction in the primary years of education.
3. Remain open to multiple modalities of different learners.
4. Develop a mindset of connections.
5. Integrate complementary concepts and procedures.

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6. Choose numbers deliberately to allow easy access.
7. Scaffold to give access and build independence.
8. Affirm students' strengths and address misconceptions.
9. Explicitly teach math language with visuals.
10. Provide professional development to support teachers in this process. (Ballin et
al., (2022)
As with any effective instruction, a commitment to student engagement and realworld functionality is essential for creating a mindset of connectivity. This should not be
lost when providing the modifications, adaptations, and intervention strategies necessary
to meet the needs of students with learning disabilities. It is critical to maintain the
perceived value of learning if long-term growth and achievement are to be sustained.
Misconceptions and Inadequate Training
One of the most overlooked aspects of special education is the lack of training
that teachers receive in the area of individualized student services (Hutchison, 2018).
Considering the onset, the average aspiring teacher has had little exposure to special
education in their academic career. It is often an eye-opening experience when these
individuals first encounter a clinical experience in which special education students are
integrated into the general education classroom and supporting teachers share their
responsibilities of following individualized education plans and 504 agreements.
Hutchison (2018) states:
Pre-service teachers often need their tools of the trade sharpened and periodically
recalibrated in order to be capable of registering true reality, as opposed to their
culturally- or societally-induced realities...because in classrooms all over the

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world, the lives of millions of real students depend on, and are determined by, the
perceptions of teachers who often use mids-calibrated instruments to determine
their psychological, social, and economic futures. (p. 113)
For this reason, it is critical that the development of young teachers and the continued
education of current teachers include an extensive focus on the practices and strategies
necessary to address the learning of students with disabilities and the understanding of
individualized needs.
Standards-Aligned Curriculum
The importance of a standardized-aligned curriculum that is guaranteed and viable
must not be overlooked when considering special education. Blank and Smithson (2014)
conducted a research study that included approximately 300 teachers in 50 sample
schools across three states in the U.S. The focus was to examine the extent to which each
school's curriculum was aligned to the standards required and the fidelity by which each
school's instruction aligned to the intended curriculum. The results indicated that at the
middle school level, instruction reported by teachers for both general and special
education students:
Did not closely align to state content standards, both in the distribution of
instructional time by topic and in the expectations of learning that are
emphasized. However, the analysis did show that a greater degree of instructional
alignment to standards did have a positive impact on student achievement...which
hold for both regular standards and extended standards for students with
disabilities. (p. 143)

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Additionally, schools that provided inclusive practices for special education students had
a positive impact on those students; however, the majority of schools analyzed identified
instructional practices in which students with disabilities received less instructional time
and more time on test preparation. The result of this was lower achievement on state
standardized tests.
Role and Importance of the IEP Team
Even if a school system is in place to provide a standards-based curriculum to all
students, inclusive of those with disabilities, there is essential work that must be
addressed by those responsible for the education of students in need. Members of the IEP
team must examine the general education curriculum to determine what is expected of
students and to determine if IEP goals can be constructed to ensure that each student can
eventually demonstrate proficiency in those areas. Present levels of academic
achievement and functional performance must also be established in order to determine
necessary interventions to address gaps in learning. If these items can be addressed
appropriately, students with disabilities can benefit significantly from inclusion in the
general education classroom (Nolet & McLaughlin, 2005).
The impact of a student's disability on achievement is often obvious and easily
addressed; however, "one of the most persistent problems that IEP teams face is that
often it is very difficult to separate the direct impact of disability from other factors that
impact school performance, such as language, class, previous educational opportunity,
culture, or various family factors" (Nolet & McLaughlin, 2005). As a result of this, IEP
teams have been encouraged to focus on students' response to intervention (RTI) in the

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general education classroom when determining if an evaluation is necessary for special
education services.
Determining a special education student's progress in the regular education
classroom is most effective when using an individual referencing assessment. Rather than
comparing each individual student to the progress of other students in the classroom
(norm referencing), it is more meaningful to monitor the progress that the student has
made over an educational period.
Individual-referenced decision making involves systematic comparisons of
students' current work with their previous performance. Individual-referenced
evaluation often is referred to as 'formative evaluation' because the effects of
instruction are evaluated on an ongoing basis rather than after all instruction has
been delivered. (Novlet & McLaughlin, 2005, p. 70)
When this is done effectively, members of the IEP team can determine if the student is
making progress and, if not, what changes and/or interventions are necessary to produce
the desired educational outcome.
Economically Disadvantaged Students
Each fall, Pennsylvania teachers and school leaders finally have the opportunity to
review student growth on state-standardized mathematics assessments from the previous
academic year. It is this time of year that the Pennsylvania Value-Added Assessment
System (PVAAS) releases growth data for each student and school personnel can easily
review each student’s progress and the success of the previous year’s instruction on
subgroups of students such as minority, special education, and economically
disadvantaged status.

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The economically disadvantaged subgroup consists of students who qualify for
the federal free and reduced lunch program as a result of their family income level. The
growth of economically disadvantaged students is a major consideration for teachers and
administrators, and it is often used as a measuring stick for the overall effectiveness of
the school’s instructional programs. These students, along with students in the
aforementioned subgroups, are the individuals in most need of the resources and
instructional expertise that our schools provide. Moderate to high levels of growth
evidence in one or more of these subgroups often correlates with a successful academic
year from a leadership perspective.
Family Income and Academic Achievement
Brooks-Gunn and Duncan (1997) concluded that family income has a significant
impact on the well-being of children. This factor significantly affects ability and
achievement rather than emotional development and stability. There is a distinct
correlation between the timing of poverty and how this impacts educational success and
school completion. Students who experience poverty early in life have less chance of
school completion than those who encounter this situation in the latter stages of their
educational experience (Brooks-Gunn & Duncan, 1997). This data suggests that early
intervention strategies can have a strong impact on the future success of economically
disadvantaged children.
Family intervention can also be impactful in addressing the economically
disadvantaged. Although educators rarely consider home life when determining how to
address academic success because of its elusive nature, increasing parental involvement
in a child's education will likely have a significant effect on the perceptions of children

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and the value that they place on their educational experience (Brooks-Gunn & Duncan,
1997). Although this theory of family intervention is not necessarily specific to the
student, school districts frequently look at ways to engage families and provide support to
members of the community through the comprehensive planning process. Many early
intervention programs exist outside of the school district’s scope of responsibility, but the
preparedness of incoming students at the kindergarten level has a crucial impact on future
success. It is reasonable for districts to consider all factors that contribute to the wellbeing of students, and with strong evidence suggesting that early life poverty is a major
predictor of educational success, this phenomenon should not be overlooked.
A meta-analysis conducted on socioeconomic status and academic achievement
consisted of over 100,000 students, 6,800 schools, and 128 school districts (Sirin, 2005,
p. 432). The results of this study showed a moderate correlation between socioeconomic
status and academic achievement at the student and family level and a stronger
correlation between status and academic achievement at the school level. Family
economic status determines the resources that are available to children in the home and
outside of the school environment, the "social capital" necessary for children to succeed
in the school setting, and the kind of school to which students have access. School
districts with a higher socioeconomic demographic were found to have better
instructional arrangements and materials, more qualified and experienced teachers, and a
lower student/teacher ratio. Additionally, both family and school impact the quantity and
quality of relationships between school personnel and parents of economically
disadvantaged students (Sirin, 2005).

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Value Assessment
In a study conducted by Hentges et al. (2019), data supported the theory that
children of low economic status continuously make value assessments based on their
current reality and they tend to perceive learning mathematics as a costly undertaking.
This likely contributes to a lower level of student achievement in this course of study.
However, economically disadvantaged students do not necessarily have less interest in
mathematics or the belief that mathematics is unimportant. Traditional instructional
methods for mathematics that do not adhere to the empowerment math philosophy often
rely heavily on “skill and drill” for math practice (Ballin et al., 2022). Often this practice
takes on the form of frequent and lengthy homework assignments so that students can get
additional application practice outside of the school day. In these scenarios, when
students have difficulty understanding mathematical concepts, completing homework can
be a more than tedious task that requires substantial time and effort with minimal results
(Ballin et al., 2022). For this reason, specific interventions that target value perceptions
and unguided practice may be the key to engaging underprivileged students in the
mathematical classroom and lead to increased overall achievement.
Expectancy Value Theory (EVT) generally refers to the expectancy of successful
completion of a task based on the perceived value of the task itself. Guo et al. (2015)
attempted to use this theory to examine how variables such as gender and socioeconomic
status predict self-concept values and task values. The belief was that these two values
have a direct impact on an individual's mathematical achievement and educational
ambition. A sample of over 5,000 8th-grade students was surveyed using a studentbackground questionnaire in which motivational items on the survey were answered

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using a four-point Likert scale. The data produced suggested a strong correlation
between socioeconomic status and educational aspirations. Students of higher socioeconomic status had higher expectations for themselves from an educational perspective,
and students of lower socio-economic status had lower standards. There was also
significant data to suggest that lower socioeconomic status had more of a negative impact
on male students than it did on the female gender (Guo et al., 2015).
Compulsory School Attendance
Pennsylvania mandates that every child between the ages of 6 and 18 must
comply with compulsory attendance requirements. Compulsory attendance refers to the
mandate that all school-aged children having a legal residence in Pennsylvania must
attend a day school in which the subjects and activities prescribed by the standards of the
State Board of Education are taught in the English language, except in situations outlined
in the Pennsylvania Public School Code (Pennsylvania Department of Education, 2023b).
It is generally understood from a legislative and operational standpoint that the presence
of students in school is necessary for sustained academic achievement and successful
completion of the secondary school experience.
In the study by Morrissey et al. (2014), it was noted that absenteeism is a
recurrent characteristic of low-income students that significantly contributes to academic
difficulty. This is a common problem that has been identified by schools, teachers, and
district leaders and has led to intervention programs aimed at increasing attendance for all
students. There is a direct correlation between time devoted to learning and the exposure
of students to the curriculum and instructional practices that expand their knowledge and
problem-solving capacity. Although the data in this study was inconclusive, it is

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reasonable to expect that lower rates of attendance among economically disadvantaged
students will contribute to a lack of achievement. Particularly in the mathematics
discipline, many applications and processes that are introduced build upon previous
knowledge and prerequisite skills. When a child misses school, it is often a priority,
especially in mathematics, to provide supplemental instruction to compensate for the time
lost and backfill the information that was unobtained by the student during the absence.
This becomes a compounding problem when absences are frequent and/or consecutive.
Getting students to the school, supplying them with the basic resources to be
healthy, alert, and attentive, providing for their individualized educational needs, and
knowing their strengths and values, are the main identified components for the success of
economically disadvantaged students (Morrissey et al., 2014). Although it may seem
trivial, these identified elements are the first step to ensuring educational equality for
those who are underprivileged.
Mixed Results and the Need for Future Study
The conclusions of the research studies in this literature review do not provide a
definitive direction for the school leader who desires to make research-based decisions to
create the most effective school system and provide the best instructional strategies for
the achievement of all students in the mid-level mathematics discipline. While this can
create a sense of frustration as school leaders face the increasing demands of government
mandates to increase student achievement in a relatively short period of time, there is
value in following the leads created by previous researchers and narrowing the focus to a
particular area of need.

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In Pennsylvania, proficiency scores on state standardized math exams
significantly decline from grade four to grade five in the vast majority of schools across
the Commonwealth. This trend continues in subsequent years until students cease
participation in the Pennsylvania System of School Assessments (PSSAs) and move on to
the Algebra I Keystone exams. Furthermore, mid-level mathematical skills are essential
for application beyond secondary education and for practical use in job-related fields.
With a growing number of STEM-related jobs and careers in the United States, it is
crucial to overcome this lapse in mathematical achievement and reset the continuous
cycle of improvement with a commitment to mathematical growth and proficiency
moving forward.
Summary
This chapter began with an introduction outlining the various reform initiatives by
the government that have resulted in frustration for school teachers and leaders who are
tasked with increasing student achievement. Although this is an interest in all core
content subjects, the English Language Arts, Mathematics, and Science disciplines are of
particular concern to district leaders because schools are formally evaluated on the
performance of their students on state-mandated standardized assessments. Researchbased pedagogical practices and knowledge of developmental paths combined with the
appropriate use of guidance, structure, and feedback are viable methods for teachers to
provide students with the best possible instruction that schools have to offer. Secondly, a
commitment to professional learning communities inclusive of the daily collaboration of
teachers and instructional teams, along with meaningful and ongoing professional
development, can support a guaranteed and viable curriculum for all students. Third, a

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comprehensive understanding of the needs of students with IEPs and the adherence to
high learning standards for every student can maximize academic growth regardless of
disability. Finally, a concerted effort to provide underprivileged students with resources
and help them find value in the daily learning that occurs within our schools will greatly
contribute to closing the achievement gap and overcoming the statistical probabilities of
future economic success. A commitment to addressing these four critical areas in the
education profession will have a substantial impact on the achievement rates of students
in a standards-based educational system.

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39

CHAPTER III
Methodology
This chapter describes the design and methods of data collection used to identify
high-achieving schools in the area of 5th-grade mathematics proficiency on state
standardized tests and determine what school systems and instructional practices are most
attributed to success. Public data was obtained and organized to identify elite groups of
schools, and demographic information was referenced to select individual schools that
most aligned to the researcher's home district. A series of structured interview questions
were then created based on the literature review to support the collection of qualitative
data from three high-achieving schools. The data was analyzed to determine common
themes, practices, and systems that school leaders implement and attribute to high-levels
of student proficiency on standardized mathematics assessments.
Purpose
The purpose of this study was to examine the relationship between school systems
and instructional methods and their impact on student achievement on state standardized
tests for mathematics as measured by the Grade 5 Pennsylvania System of School
Assessment (PSSA) in the Commonwealth of Pennsylvania. Further, this study analyzed
the impact on the achievement of students identified as economically disadvantaged or
having identified learning disabilities. This research is relevant as school districts in
Pennsylvania are searching for meaningful systems and strategies to increase student
achievement in the area of mid-level mathematics.
The research method utilized in this study was a mixed method strategy aimed to
investigate factors influencing 5th-grade mathematics proficiency in Pennsylvania by
combining quantitative data from the Pennsylvania Department of Education with

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qualitative insights gathered through interviews with principals from three high-achieving
schools in the state. The quantitative analysis involved examining state standardized test
scores and demographic information to identify high-achieving schools in the area of
mathematics among 5th-grade students. Concurrently, qualitative interviews with the
principals and leaders of these schools explored perceptions of effective teaching
practices, curriculum design, student support systems, and overall school culture that
influences high levels of mathematical learning. By triangulating these data sources, the
study sought to provide a comprehensive understanding of the factors contributing to
mathematics proficiency and offer insights into potential strategies for improvement.
Mathematics proficiency is a critical aspect of a student's education, influencing
their academic success and future career opportunities. Recognizing the significance of
this, educational institutions constantly seek effective strategies to improve math
proficiency among their students. This research project aims to investigate the school
systems and instructional strategies implemented by building principals and school
leaders to enhance student math proficiency.
This study is relevant to the researcher because mathematics proficiency on
standardized tests have been historically low at the 5th-grade level in the Kiski Area
School District. This is coupled with the fact that the most significant drop across grade
levels historically occurs from the 4th grade level to 5th grade level. The decreased rate of
proficiency continues in subsequent grades until the 8th grade year when students are no
longer required to take the Pennsylvania System of Student Assessment (PSSA) exams.
Table 1 illustrates mathematical proficiency rates in grades 3-8 in the Kiski Area School
District on the 2021-2022 PSSA Math exams.

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Table 1
Kiski Area School District PSSA Math Proficiency by Grade Level, 2021-2022

Note. Table 1 shows the largest difference in proficient and advanced scores on the PSSA
Math exams occurred between the 4th-grade cohort and the 5th-grade cohort of students.
The percentage change in overall proficiency for the 2021-2022 school year was 36.41%.
This negative difference continued in grades 6, 7, and 8, but at a much lesser rate.
The Kiski Area School District is made up of three (3) K-4 Primary Schools, one
(1) 5-6 Upper Elementary School, and one (1) 7-8 Intermediate School. It has a total
district enrollment of approximately 3,384 students, and has a geographic size of 104.96
square miles.
It was noted throughout this research project that this district structure is unique
when compared to other district structures across the commonwealth of Pennsylvania.
The majority of school districts observed in the research process implemented a system in
which 5th-grade students were incorporated at the primary level. Middle school models
tended to incorporate 6th-grade students in their youngest grade level.
Setting and Participants
The setting of this research study was determined by the demographic make-up of
the researcher’s home school district. It was essential to identify specific data points such

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42

as total student population, percentage of economically disadvantaged students, and
percentage of students with learning disabilities in order to compare student achievement
across districts and school entities in a meaningful manner. When high-achieving schools
were identified using the yearly results published on the Pennsylvania Department of
Education website, the Future Ready PA Index was utilized to identify the demographic
make-up of those schools and determine if enough similarity existed to classify those
schools as candidates for further research. Table 2 illustrates the percentage of students
enrolled in the Kiski Area Upper Elementary School by state-identified subgroups.
Table 2
Kiski Area Upper Elementary School Percent Enrollment by Student Groups, 2023-2024
(Future Ready PA Index, 2023a)

Note. Table 2 shows the significant percentage of students enrolled at Kiski Area Upper
Elementary School who are considered economically disadvantaged (49.9%), and the
percentage of students receiving special education services at 15.8%.

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It was determined by the researcher that the process of school identification
should be limited to three (3) specific schools with comparable demographic data to that
of the researcher’s home district and school. The reason for this was to ensure an in-depth
analysis of qualitative data obtained from a limited number of schools rather than a less
comprehensive examination of a larger entity sample. It was also determined that meeting
demographic parameters set by the researcher significantly limited eligible schools when
considering a minimum level of mathematical proficiency that schools must exhibit. This
led to the creation of specific criteria that was used to categorize and sort schools that
correlated most with the researcher’s home district.
The process of school identification was based on the following criteria: 1) The
school must service students enrolled in 5th grade. 2) The school must serve an
economically disadvantaged population that exceeds 20% of its enrollment. 3) 5th-grade
mathematics proficiency must exceed 60% on the 2022-2023 state standardized
mathematics assessment. These conditions led to the selection of three elementary
schools in the state of Pennsylvania that became the target of qualitative research in this
study.
Research was conducted in three elementary schools from three different school
districts in Pennsylvania. The subjects of this project were building principals or district
leaders who are directly involved in the leadership of student learning in the selected
entities. The researcher submitted a plan to each district’s superintendent and obtained a
written letter of approval to conduct research in the identified schools.
Based on the literature reviewed in Chapter II, student achievement in
mathematics is highly impacted by instructional methods and systematic processes that

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can vary greatly from one educational organization to another. In order to explore these
phenomena across multiple organizations, specific questions were developed that align
the information obtained in the literature review process as well as the research questions
that are the focus of this study. These questions made up a semi-structured interview
proposal that was submitted to Internal Review Board (IRB) of Penn West University and
approved for implementation in September of 2023.
The first school (School A) in this study was a K-5 elementary school located in
Southwestern Pennsylvania. Although the district in which this school is located has a
significantly different socio-economic population when compared to that of the Kiski
Area School District, the geographic setting, overall student population, and the
percentage of special education students serviced made this school a valid selection,
especially considering its historic success in exceeding mathematics proficiency
standards at the 5th grade level. School A’s district consists of five elementary schools
(grades K-5), two middle schools (grades 6-8), and one high school (grades 9-12). It has
a total district enrollment of approximately 3,879 students, and has a geographic size of
34.02 square miles. Table 3 illustrates the percentage of students enrolled in School A by
state-identified subgroups.

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Table 3
School A Percent Enrollment by Student Groups, 2023-2024 ((Future Ready PA Index,
2023b)

Note. Table 3 shows the percentage of students enrolled at School A that are considered
economically disadvantaged is 22.5%, and the percentage of students receiving special
education services is 13.6%.
The second school (School B) that was identified for the purposes of this study
was another K-5 elementary school located in Western Pennsylvania. Although this
school had a significantly smaller population when compared to the Kiski Area Upper
Elementary School, the state-identified student groups aligned very closely with that of
the researcher’s home school, and the geographic setting was also very similar. School
B’s district consists of one elementary school (grades K-5), one middle school (grades 68), and one high school (grades 9-12). It has a total district enrollment of approximately
713 students, and has a geographic size of 11.95 square miles. Table 4 illustrates the
percentage of students enrolled in School B by state-identified subgroups.

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Table 4
School B Percent Enrollment by Student Groups, 2023-2024 (Future Ready PA Index,
2023c)

Note. Table 4 shows the percentage of students enrolled at School B that are considered
economically disadvantaged is 66.0%, and the percentage of students receiving special
education services is 16.6%.
The third and final school (School C) that was identified in this research project
was a 2-5 elementary school located in Southeastern Pennsylvania. School C’s
geographic make-up is the main outlier when compared to Kiski Area Upper Elementary
School because it is in a more urban setting; however, socio-economic and special
education data aligned effectively with that of the researcher’s home district. School C’s
district consists of one literacy center (grades K-1), one elementary school (grades 2-5),
one middle school (grades 6-8), and one high school (grades 9-12). It has a total district
enrollment of approximately 4,390 students, and has a geographic size of 6.7 square

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47

miles. Table 5 illustrates the percentage of students enrolled in School C by stateidentified subgroups.
Table 5
School C Percent Enrollment by Student Groups, 2023-2024 (Future Ready PA Index,
2023d)

Note. Table 5 shows the percentage of students enrolled at School C that are considered
economically disadvantaged is 29.1%, and the percentage of students receiving special
education services is 18.1%.
Research Plan
The researcher utilized a mixed method approach to complete the research
outlined in this doctoral capstone project and address the identified research questions.
The researcher submitted a plan to the Internal Review Board (IRB) of PennWest
University and this plan was accepted and approved on September 14, 2023 (Appendix
A). The quantitative research in this project involved the collection of state assessment
data from the Pennsylvania Department of Education website to identify grade-level

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proficiency rates in mathematics proficiency, and determine the most significant change
in student achievement across grade levels. Table 6 illustrates a comprehensive view of
mathematical proficiency on state standardized math assessments across all schools in the
state of Pennsylvania that participated in these assessments during the 2021-2022 school
year.
Table 6
2022 PSSA Math Results, Grades 3-8 (Pennsylvania, 2023)

It was observed by the researcher that the 2021-2022 results showed the most
significant decrease in proficiency across grade levels occurring at the 5th grade level.
Although the last column in the table shows a pattern of decreasing proficiency from each
grade level to the next, the 6.9% decrease in the percentage of proficient and advanced
students from grade 4 to grade 5 is the most extensive in this data set.
It was also observed at the researcher’s home district level that mathematical
proficiency at the 5th grade level for state standardized tests has been historically dismal
and stagnant over a five-year period of time. Table 7 illustrates the percentage of students
proficient and advanced on the yearly PSSA exams from the 2017-2018 school year to
the 2022-2023 school year.

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Table 7
Kiski Area Upper Elementary Math PSSA Results Since 2018

Note. Table 7 also shows how the Kiski Area Upper Elementary School breaks down
proficiency by each reporting category present on the standardized exam for the 20222023 school year. Highlighted values show the tested year along with the overall
proficiency rate for that year. It is also notable that a 32% overall proficiency rate was
recorded for the 2022 testing year. This falls below the 35.4% average proficiency rate
of all 5th grade students in the commonwealth of Pennsylvania that were administered the
exam in 2022.
The quantitative data obtained in this process led the researcher to develop
research questions surrounding the phenomenon of decreased mathematical proficiency
at the 5th-grade level, not only from the comprehensive perspective of the state of
Pennsylvania, but also locally within the researcher’s home district. Furthermore, the
demographic make-up of the researcher’s home district contributed to the need for
additional inquiry based on a significant percentage of students making up the
economically disadvantaged and students with learning disabilities subgroups within the
targeted school. The research questions established as a result of the review of literature
and the quantitative analysis are:

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Research Question 1
What instructional strategies and methods do high-performing schools employ to
achieve high proficiency rates in mathematics among all students?
Research Question 2
What instructional strategies and methods do high-performing schools employ to
achieve high proficiency rates in mathematics among students who are economically
disadvantaged?
Research Question 3
What instructional strategies and methods do high-performing schools employ to
achieve high proficiency rates in mathematics among students with learning disabilities?
The final quantitative approach used in the research process was implemented to
determine which three (3) schools in the state of Pennsylvania significantly exceeded the
average proficiency standards documented by the Pennsylvania Department of Education
for the 2021-2022 school year as outlined in Table 6. This was accomplished by
downloading the “2022 PSSA State Level Data” spreadsheet that is public and made
available on the Pennsylvania Department of Education Website (Pennsylvania
Department of Education, 2023c). The data showed the results of all standardized testing
for every public school in the state of Pennsylvania that participated in the Pennsylvania
System of Student Assessments (PSSAs) for the 2021-2022 school year, and included
more than 1,400 schools across the Commonwealth. This spreadsheet was sorted by
grade, subject tested, percent proficient overall, percent proficient economically
disadvantaged, percent proficient special education, and district name to determine highperforming schools based on those criteria. The PSSA State Level Data spreadsheet was

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51

also valuable in organizing data to determine high-achieving schools among students
with low socio-economic status and students with learning disabilities.
Once three schools were selected based on their high levels of achievement in
mathematics and similar demographic make-up to that of the researcher’s home school
and district, the next step was to recruit administrative participants from these schools for
participation in a semi-structured interview. The individuals identified in this process
were principals and school or district leaders directly involved in the implementation of
math curriculum and instruction. The organization and specific questions used in the
interview are described in detail in the next section of the methodology, but the main
purpose of the interview was to obtain data from each school related to mathematical
systems and instructional practices that lead to high levels of student achievement on
math standardized assessments. Because the researcher’s home school and district
achieved results below the state average for mathematical proficiency on the 2021-2022
PSSAs starting in grade five (5), the main goal was to identify similar strategies,
programs, and overall operations used by successful schools that could be implemented
by the Kiski Area School District to improve student achievement. Success in this area
will be measured by a tangible and steady increase in the percentage of students who
achieve an advanced or proficient score on the PSSAs in future testing years.
Methods of Data Collection
The method of data collection involved a comprehensive process for obtaining
permission to conduct educational research across three (3) distinct school districts in
Pennsylvania. As referenced throughout this section, the three (3) schools identified were
selected based on their enrollment of 5th-grade students, a population of economically

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52

disadvantaged students that exceeds 20% of this enrollment, and an overall proficiency
rate of 60% or higher on the 2022-2023 PSSAs.
On September 5, 2023, formal requests for permission to conduct research were
emailed directly to the superintendents of each respective district. Each superintendent
granted permission for the research to occur in his district, and on September 11, 2023,
the researcher submitted formal letters of approval provided by those superintendents to
the PennWest Internal Review Board (IRB) for approval. After securing the necessary
approval, building principals of three identified schools within the approved districts
were contacted via email to solicit their participation in the research study (Appendix B).
This methodical approach ensured adherence to ethical guidelines, proper authorization
from relevant authorities, and systematic participant recruitment.
Once contact was established with each identified building principal/school
leader, an IRB approved consent form was emailed to each participant for completion,
along with the structured interview questions that would be presented at an agreed-upon
meeting date and time (Appendix F). The principal of School A in this study returned the
signed consent form on February 23, 2024, and the semi-structured interview took place
using a Google Meeting platform on March 1, 2024 (Appendix C). The interview lasted
approximately 42 minutes and was recorded both visually and auditorily, as well as
transcribed using the Google Meeting transcription tool.
The principal of School B returned his consent form the morning of March 25,
2024, and the semi-structured interview occurred using the same aforementioned Google
Meeting platform on the afternoon of March 25, 2024 (Appendix D). The interview

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lasted approximately 47 minutes, and was also recorded visually and auditorily, and
transcribed using the Google virtual meeting platform.
After three attempts to establish a connection with the principal of School C via
the email recruitment letter, I received a response from this individual on March 23,
2024. Due to unforeseen family circumstances, the principal directed me to the district’s
director of curriculum and instruction, as she believed this leader could more than
adequately provide responses to the interview questions that were provided in relation to
her assigned school. After several attempts to connect with the district’s director of
curriculum and instruction, he finally returned my email and further directed me to the
district’s Title 1 coordinator who was a former teacher at School C and heavily involved
in the shared leadership of that building. The formal recruitment letter, along with a
description of the circumstances for a change of contact, was sent to the district’s Title 1
Coordinator on May 15, 2024. This individual returned her consent form on May 16, and
a semi-structured interview occurred using the Google meeting platform on the morning
of May 17, 2024 (Appendix E). The interview lasted approximately 41 minutes, and was
recorded visually and auditorily, and transcribed using the Google virtual meeting
platform.
The structured interview questions developed in this research project (Appendix
F) were directly formulated from the results of the literature review process outlined in
Chapter 2. The qualitative questions presented were categorized into five (5) main
sections. These sections included Special Education Students, Economically
Disadvantaged Students, External Factors, School Services and Resources, and General
questions for school and participant characteristics. Although the questions were created

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and provided to the participant prior to the actual interview itself, the interview process
was semi-structured. All questions were asked by the researcher and answered by the
participant, but additional clarifying questions were posed when necessary, and dialog
occurred based on many of the responses that were provided. There were several
responses to questions that provided insight, or at times a complete narrative, that was
able to be used to achieve data for subsequent questions.
The three schools selected by the researcher using this filtering method were not
the schools at the top of each list; however, the chosen schools were those that had both
high mathematical achievement and a similar demographic population to the researcher’s
home district and school. Specifically, one of the schools selected in this study ranked in
the top fifteen (15) of the overall standings of 5th-grade proficiency on the 2022 PSSA
math exams with a total proficiency rate exceeding 85%. This school did not, however,
break the top 50% of schools with high achieving economically disadvantaged and
special education populations.
On the other hand, another school selected ranked relatively high in all three data
sets reviewed. It was in the top five (5) in Pennsylvania among schools with a highachieving subgroup of economically disadvantaged students at a proficiency rate which
exceeded 74%. It ranked in the top twenty-five (25) among schools with a highachieving special education subgroup exceeding a proficiency rate of 48%, and in the top
20 in the overall category of schools surpassing an 82% advanced/proficiency rate.
The final school selected was ranked just inside the top one-hundred (100) schools
for 5th-grade proficiency on the 2022 PSSAs, but its economically disadvantaged
population ranked in the top twenty (20) in Pennsylvania with a subgroup proficiency

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rate surpassing 62%. School B also fell just inside the top 50% of schools with a highachieving special education subgroup.
The qualitative portion of the mixed method research plan focused on the
questions that were produced to conduct a semi-structured interview with leaders of three
(3) high-achieving schools on 5th-grade standardized mathematics assessments. The
creation of these questions was highly influenced by the literature review that was
conducted by the researcher and outlined in Chapter II of the capstone project.
The interview was organized into five (5) designated categories and consisted of
twenty-two (22) total questions (Appendix F). The first section was titled “General
Questions” and consisted of four (4) questions, the second section was titled “Special
Education Students” and contained five (5) questions, “Economically Disadvantaged
Students” was the middle section of questioning making up five (5) questions, and the
final two sections were “External Factors” and “School Services/Resources,” consisting
of three (3) questions and five (5) questions respectively.
In the first of a series of semi-structured interviews, the researcher engaged in a
comprehensive discussion with the principal of a K-5 elementary school located in
western Pennsylvania. This is referred to a “School A” throughout the research project.
With a tenure spanning 17 years at the school, the principal brought a wealth of
experience, having previously served as a high school math teacher. The interview
provided insights into the school's notable success in consistently surpassing math
proficiency standards within the region at the 5th-grade level.
In the second interview conducted as part of the research project, the researcher
engaged with the principal of a school servicing students in grades 2-5. This is referred to

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as “School B” throughout this research project. This school leader brought a unique
perspective to the discussion, having transitioned from a role as a high school assistant
principal to assume leadership at the elementary level. With a background as a special
education teacher at the secondary level spanning 12 years, the principal offered insights
into the school's approach to achieving mathematical proficiency, particularly among
economically disadvantaged students. At the time of this conversation, it was noted that
29% of the school’s student body belonged to the low socioeconomic subgroup.
In the final interview conducted as part of the qualitative analysis, the researcher
met with the Title I Coordinator that services “School C”, a K-5 elementary school also
located in Western Pennsylvania. Due to extenuating circumstances, the principal of this
school was unable to formally participate in the study; however, the researcher was
ensured by this individual that the Title I Coordinator had an intimate understanding of
the systems and operations in place within this entity, and would be the most
knowledgeable candidate for the purpose of the research being conducted. Nonetheless,
this district leader was able to provide a more system-rich perspective of 5th-grade
mathematics achievement because of her role servicing students from all grades
kindergarten through twelve (12).
At the conclusion of each conducted interview, responses to the questions
provided were organized into an Excel spreadsheet to compare and contrast the data
provided by each building leader. The researcher used a color-coding structure to
highlight and categorize similar systems and methods referenced by each participant. The
purpose of this categorization structure was to find consistently targeted areas among the
three schools and gauge the level of commitment each district allocates to a particular

57

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system or methodology. Table 8 illustrates the method used by the researcher to analyze
data and identify themes and commonalities among school leader responses.
Table 8
Foundational Categories of School Systems and Methodologies
Foundational Category

System/Methodology
Vertical Alignment

Curriculum

Horizontal Alignment
Curriculum Development/Approach
Assessment Strategy

Instruction/Assessment

Instructional Strategy
Lesson Planning Strategy
Professional Development

Professional Learning

Collaborative Approach
Coaching Strategy
Human Resource

Resources

Textual Resource
Curriculum Resource
Parent Resource
Growth Strategy

Student Learning

Retention Strategy
Intervention Strategy
Engagement Strategy

Note. Table 8 identifies a general foundational category that is then broken down into
more specific systems and methodologies based on the participant’s response.
Fiscal implications of this research project were minimal. The research plan
involved significant time commitments by the researcher to analyze state assessment
data, develop structured interview questions, recruit participants, and organize online,

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virtual interviews; however, minimal time was required by each participant to prepare for
and participate in the interview process. It was estimated that each interview would take
between 45-60 minutes to complete, and each interview fell within that time range.
The costs associated with recommended changes and improvements to the
researcher’s home district based on findings are, however, outlined in Chapter V of this
research study. In order to achieve meaningful and sustainable results in the area of
mathematics proficiency on standardized assessments, funds must be allocated to
replicate school systems and instructional practices that are correlated most consistently
with academic success.
Validity
As outlined in the consent to participate in the research study created by the
researcher and approved by the PennWest IRB (Appendices C, D, and E), participating
school leaders in this project were required to service 5th-grade students, have an
economically disadvantaged population exceeding 20% of their total school population,
and have an overall proficiency rate of 60% or higher on state standardized mathematics
assessments for the 2021-2022 school year. The purpose of this requirement was to
ensure transferability, which is the degree to which research results are applicable to
other contexts and other individuals (Hendricks, 2017). It was important to analyze
comparable educational settings in order to transfer findings and results into application
in the researcher’s home district.
It was also outlined in the consent form that participation in the interview process
was voluntary and the participant had the right to refuse to answer any question or
withdraw any response after the process was completed. A transcript of each interview

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was created and provided to the participant upon request for review. All personally
identifiable information provided in the interview process was redacted, and participants were
informed that names would never appear on research instruments or in the capstone manuscript.
Participants were informed that all written and electronic forms and study materials, including
audio and video recordings, would be kept secure and password protected, and that any study
materials with personal identifying information will be maintained for three years after the
completion of the research and then destroyed. The purpose of this structure was to ensure
credibility. The researcher utilized this method in an attempt to ensure that interview results were
accurate and truthful, and not a scrutiny of specific strategies and techniques supported and
implemented by the subject of the interview.

Finally, the researcher attempted to establish credibility, dependability, and
confirmability by implementing the process of triangulation (Hendricks, 2017). The
research triangulation process began by gathering 5th-grade student math proficiency
data from the Pennsylvania System of School Assessments (PSSAs) for the 2021-2022
school year. Three schools were then strategically selected based on their high-levels of
student proficiency and comparable demographics to the researcher's home district.
Through semi-structured interviews, school leaders from each of the chosen schools
provided qualitative insights into the systems and practices they attributed to their
success. This multi-faceted approach ensured a comprehensive understanding of factors
contributing to student achievement, incorporating both quantitative proficiency metrics
and qualitative perspectives from educational leaders, thus enriching the depth and
validity of the research findings.
Furthermore, the researcher cross-analyzed results from each selected school to
find commonalities among programs, systems, and instructional practices that school

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leaders have directly credited to high levels of student achievement. Although it was rare
to find direct similarities between specific programs and practices utilized by the schools
involved in the study, parallel foundational theories and methodologies were derived
through inquiry and discussion in the semi-structured interview process. This additional
method of triangulation further supported tangible approaches in results-driven
educational entities.
Summary
The purpose of this Methodology chapter was to outline the purpose, setting,
research plan, data collection process, and validity of the overall research study. Based on
a historical and ongoing lack of student achievement in mid-level mathematics in the
researcher’s home district, quantitative data was obtained to show a similar trend across
many districts in Pennsylvania. The researcher used this data to focus the study on the
5th-grade level, as it statistically showed a defining point in the trend of declining
mathematics proficiency, both in the target school and across other schools in the
commonwealth of Pennsylvania.
The Literature Review chapter provided invaluable insight on systems and
practices utilized for mathematical engagement and achievement, and directly contributed
to the formulation of an interview process that addressed instructional practices,
demographic make-ups of schools, practices associated with population subgroups, and
school systems that contribute to high levels of student learning.
The process concluded with the selection of three (3) target schools that were
identified based on their mathematical proficiency on state standardized assessments,
specifically considering the performance of their economically disadvantaged student

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population, their special education population, and their overall student enrollment.
School principals and district leaders were then interviewed utilizing the questions
created as a result of the literature review process in an effort to determine the
mathematical systems, instructional practices, and other educational factors that led to
student success in mathematical operations and applications.
The next chapter will analyze the data and outcomes of this research and
explicitly address the findings associated with each of the three research questions
outlined in the study. Specific components outlined in this methodology chapter will be
highlighted and expanded-upon in the in-depth analysis of results to follow.

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CHAPTER IV
Data Analysis and Results
Mid-level mathematics plays an instrumental role in the overall academic success
of today’s students. The Pennsylvania Department of Education (PDE) outlines specific
reporting categories in grades 3 through 8 consisting of numbers and operations,
algebraic concepts, geometry, and data analysis and probability. Mastery of these
mathematical concepts not only serves as a foundation for advanced mathematical
studies, but also enhances critical thinking and problem-solving skills, which are vital
across all disciplines. In Pennsylvania, proficiency in mid-level mathematics is closely
linked to academic achievement, standardized test performance, and college readiness.
Additionally, a solid understanding of these mathematical concepts is essential for
students aspiring to pursue careers in science, technology, engineering, and mathematics
(STEM) fields, which are increasingly important in our nation’s economy. By prioritizing
mid-level mathematics education, Pennsylvania can ensure its students are well-prepared
to meet the demands of higher education and the modern workforce.
In Pennsylvania, standardized mathematics scores consistently fall below the state
standard beginning at the 5th-grade level, with a noticeable trend of decreased
proficiency rates in subsequent years. This decline was most significantly pronounced
when comparing the 4th-grade cohort to the 5th-grade cohort at the conclusion of the
2021-2022 school year. The drop in proficiency between these two grades was not only
apparent at the state level, but also at the local level in the Kiski Area School District,
signaling a critical point where students' grasp of mathematical concepts significantly
weakens. This pattern suggests a crucial need for intervention and support at this level to
address and reverse the declining trend in math proficiency among Pennsylvania students.

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The purpose of this qualitative research study was to develop an understanding of
instructional practices and educational systems through the shared experiences of school
leaders in Pennsylvania public schools. The focus was on schools with an economically
disadvantaged population exceeding 20% of their student population and a proficiency
rate at or above 60% at the 5th-grade level as measured by the annual Pennsylvania
System of School Assessment (PSSA) for mathematics during the 2021-2022 school
year.
This chapter will focus on the analysis of qualitative data collected through semistructured interviews with educational leaders from three different schools across the
Commonwealth of Pennsylvania. These individuals had a direct impact on the high-level
of mathematical achievement obtained at the 5th-grade level during the 2021-2022 school
year. The data will indicate the instructional practices, school systems, and academic
programming most attributed to the success of student learning in mid-level mathematics.
Data Analysis and Findings
The first interview conducted in the research process took place on March 1st,
2024 with the principal of School A. During this interview, the principal emphasized the
school's collaborative approach among teachers, particularly in the area of mathematics
education. One notable strategy highlighted was the flexible grouping of students based
on the results of trimester benchmark exams. These exams are created by math teachers
using the Pennsylvania Standards Aligned System (SAS) website, in an effort to provide
a comprehensive assessment of the standards and eligible content taught over a
designated period of time. The results of these exams are reviewed in a team
collaborative, and students are grouped so that teachers can tailor instruction to meet the

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diverse needs of students, ensuring that each child receives targeted support and
challenges appropriate to their level of proficiency. By leveraging the data gathered from
these benchmark assessments, the school fosters an environment of individualized
learning, promoting both academic growth and student engagement.
Additionally, the interview shed light on the school's data-focused culture,
emphasizing the use of assessment data to inform instructional decisions and predict
student performance on state-standardized math tests. Through careful tracking of student
growth and progress, as evidenced by the benchmark exams, the school has continued to
increase its ability to accurately predict student outcomes on standardized assessments.
This data-driven approach not only facilitates targeted interventions for struggling
students but also assists with the identification of trends and patterns that inform
curriculum planning and instructional strategies school-wide. Overall, the interview
underscored the principal's commitment to excellence in mathematics education and
provided valuable insight into the instructional practices and systems that have led to the
school's success in exceeding proficiency standards.
The second interview conducted as part of the research project took place on
March 25, 2014 with the principal of School B. This principal of highlighted the
significance of building positive relationships with families within the community as a
foundation of the school's success. By hosting numerous family engagement sessions
throughout the academic year, the school promotes a collaborative partnership between
educators and parents, creating a supportive environment that leads to a shared
responsibility of student learning. Leveraging Title I goals and available state funding,

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these sessions serve as platforms for sharing resources, providing academic support, and
strengthening ties between the school and its diverse community.
Furthermore, the principal highlighted the school's commitment to a standardized
math resource that has been consistently utilized across all grade levels for the past five
years. This continuity ensures alignment of curriculum and instructional practices,
promoting vertical curriculum alignment and consistent math instruction school-wide.
Additionally, the school integrates various online math programs, both within the school
environment and for home use, to cater to students' individual proficiency levels. By
offering differentiated learning opportunities tailored to students' needs, the school
empowers learners to build upon their mathematical skills in a personalized and engaging
manner, ultimately contributing to their academic success.
The third interview in the process took place on May 17, 2024 with a former
mathematics teacher at School C, and the current K-12 Title I Coordinator of the district
in which School C is located. This individual was highly recommended by both the
principal of School C, as well as the district’s director of curriculum. In this interview,
the coordinator emphasized the crucial role of fostering strong community relations and
ensuring the availability of educational resources for both students and parents. She
highlighted how active engagement with the community creates a supportive network
that enhances student learning and well-being. The coordinator detailed various
initiatives, such as parent workshops and community events, designed to empower
families with the tools and knowledge needed to support their children's education.
Additionally, she stressed the importance of the district’s early intervention programs
which are free to all district families and available to children at the age of 3. These

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programs are fully funded by the district and are inclusive of transportation services to
ease restrictions faced by many families within the community.
At the completion of all three (3) semi-structured interviews, the researcher
utilized a method of data organization to analyze results across building entities. An
Excel spreadsheet was created to categorize systems and methods the were consistently
referenced by all participants. This system of data review was referenced in the previous
chapter relating to research methodology (see Table 8 – Foundational Categories of
School Systems and Methodologies).
The first foundational category identified by the researcher was curriculum. This
was broken into three specific sub-categories based on responses provided by
participating school leaders. Common themes referenced within this category included
vertical curriculum alignment, horizontal curriculum alignment, and approach to
curriculum development. Table 9 illustrates the first foundational category and notable
findings across all three participating schools.

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Table 9
Curricular Systems and Methodologies
Data Findings
Foundational
Category

System/Methodology

School A

School B

School C

Vertical Alignment

K-5 Primary School
Setting; 6-8 Middle
School
Do Not Teach
Outside of GradeLevel Standards

K-1 Literacy
Center; 2-5
Primary School
Setting; 6-8
Middle School

K-5 Primary
School Setting; 68 Middle School

Horizontal Alignment

5 Primary Entities
Less Collaboration
Across Entities
Strict District
Curriculum
Timelines

Curriculum
Development/Approach

Professional
Learning
Communities
Record-Keeping
Sheets
District-Developed
Curriculum
Timelines (MathStrict)

Curriculum

1 Primary Entity
All Students
2 Primary Entities
Exposed to Same
Curriculum

enVision Math
Curriculum - 5
years of
implementation

5th-Grade
Departmentalizati
on

Note. Table 9 identifies specific data from each interview that aligns with the overall
system or methodology related to curriculum.
The second foundational category in this study was instruction and assessment.
This was divided into three sub-categories based on responses provided by the interview
participants. Common themes referenced within this category include assessment
strategies, instructional strategies/practices, and lesson planning strategies. Table 10
illustrates the instruction/assessment category and notable findings across all three
participating schools.

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Table 10
Instruction and Assessment Systems and Methodologies
Data Findings
Foundational
Category

System/
Methodology

School A

School B

Teacher-Made Trimester Benchmarks
PSSA Math Coach
(SAS Resource)
Assessment Strategy
Assessment
STAR for Reading Only
iReady Math
Benchmarks Build on
Content Taught

Instruction/
Assessment

School C

iReady Math

Instructional
Strategy

Spiral Review (Go
Full Inclusion
Full Inclusion - CoMath)
70-Minute Math Blocks
Teaching with
20-30% Special Edu
per Day
Regular Teacher
pull-out math - same
Learning Support
and Special
curriculum/small
Teachers Assist
Education Teacher
group

Lesson Planning
Strategy

Departmentalization Group Students Based
on Benchmark
Assessment Data
4-Teacher
Flexibly Group and
Departmentalization Departmentalization
Adapt Lessons Based on (5th Grade Math)
Proficiency
Students Move 3 to 4
Times Per Year

Note. Table 10 identifies specific data from each interview that aligns with the overall
system or methodology related to instruction and assessment.
The third foundational category identified was professional learning/development.
This was also divided into three sub-categories based on the responses of interview
participants. Common themes referenced within this category include professional
development, collaborative approaches, and coaching strategies. Table 11 illustrates the
professional learning category and notable findings across all three participating schools.

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Table 11
Professional Learning Systems and Methodologies
Data Findings
Foundational
Category

Professional
Learning

System/
Methodology

School A

School B

School C

Professional
Development

Math Specific PD Language for Word
Problems- Through
PDE

PSSA Math Coach
Assessment Professional
Development on
Assessment Techniques
and Data Review

Departmentalized
PD

Collaborative
Approach

Professional
Learning
Communities
All 5th-Grade
Teachers Teach
Math Groups

7 Total 5th-Grade
Teachers - 4 Teach Math
in a Block System

3 Math Teachers
per Grade

Coaching Strategy

2 Math Coaches in
the Past

Elementary Director of
Teaching and Learning Works with Teachers

Title 1 Coordinator
- Former
ELEM/MS Math
Teacher

Note. Table 11 identifies specific data from each interview that aligns with the overall
system or methodology related to professional learning.
The fourth foundational category identified was resources. This category was
divided into four sub-categories based on the responses of interview participants.
Common themes referenced within this category include human resources, textual
resources, curriculum resources, and resources for parents. Table 12 illustrates the
resources category and notable findings across all three participating schools.

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Table 12
Resource Systems and Methodologies
Data Findings
Foundational
Category

Resources

System/
Methodology

School A

School B

School C

Director of Teaching
5 Full-time Special
and Learning for
Education Teachers KElementary 5 (including life skills)
Responsible for Data 3 Math Teachers per
Review
Grade

Human Resource

SLA - Student
Learning Assistants
(2)

Textual
Resource

All Teacher Created
Believe Resources
Did Not Fit Math
Vision (Too Many
Instructional Options)
Used EnVision Math
in the Past

enVision Math

Go Math (HM)

Curriculum
Resource

Based on State
Standards and
Eligible Content by
Grade-Level

Xtra Math - For Fact
Fluency

iReady Math

Non-Title 1
Economically
Disadvantaged
Population Around
20%

Title 1 - Math/Reading
Nights 4 Times/Year
Parent/Community
Google Classroom
Engagement Through
Training
Title 1
School Activities
Ice-Cream Social
Frequent, Consistent,
Teacher-Parent
and Open
Rapport and
Communication
Communication
After-School Tutoring
- District Funded

Parent Resource

Note. Table 12 identifies specific data from each interview that aligns with the overall
system or methodology related to resources.
The fifth and final foundational category identified was student learning. This
category was divided into five sub-categories based on the responses of interview
participants. Common themes referenced within this category were growth strategies,
retention strategies, intervention strategies, special education strategies, and engagement

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strategies. Table 13 illustrates the student learning category and notable findings across
all three participating schools.
Table 13
Student Learning Systems and Methodologies
Data Findings
Found.
Category

System/
Method.

School A

School B

School C

Growth
Strategy

Full Inclusion
Flexible Grouping Based on
Benchmark Assessment
Results

Strategically Schedule
Students in Groups based
on IEP Goals/Similar
Learning Profiles for
Intervention

Inclusion with SGI
for 20-30% of SE
Students
iReady Math
Assessments

Retention
Strategy

Spiral Review - Beginning of
School Day

Xtra Math - Fact Fluency
Program (10-15 Minutes
at the Beginning of Math
Lessons

Go Math Spiral
Review

Completed Within The Math
120-Minute Block of
Classroom Using the Flexible
Additional Math
Grouping Model. Historically
Instruction/Week - Used
Low Performing Students
for Extension and/or
Participate in One-On-One
Intervention; Re-teaching
Intervention with a Learning
for IEP Goals
Support Teacher in Addition
to Their Regularly Scheduled
Title 1 Math (All
Math Class
Students) - 30 Minutes of
Intervention
Workshop Math During
Students Can Be Pulled From
Student
Strategy
Specials Classes - Once
Specials Classes or Science
Learning
Per 6-Day Cycle
and Social Studies Classes If
Needed For Math or Reading
After-School Tutoring Intervention
One Day per Week
Homework Club - One
Homework Club - MondayDay per Week
Thursday (30 Minutes) No Cost to Parents
Parents Responsible for
Transportation Not
Transportation - Teachers
Provided"
Paid By District
Full Inclusion, But
Full Inclusion
Special
Scheduling is Strategic to
Students Flex-Grouped by
Education
Group Students Based on
Benchmark Assessment
Strategy
Learning Profiles
Results
Co-Teaching Model
Use a Variety of
Teach Students How to Use
Resources to Engage
Engagement
Calculators Consistently
Students (iReady Math;
Strategy
Calculators Used to Check
Xtra Math; PSSA
Work, Not Complete
Coaching Assessment;
Targeted Intervention)

Weekly:
Wednesday,
Thursday, Friday
Math Intervention
Periods
Early Intervention 03 - Once per Month
3-Year-Old Program
Tuesdays/Thursdays
(2 Hours/day)
Transportation
provided for both
programs
After-School
Tutoring - District
Funded

Full Inclusion/2030% Small Group
Instruction (Pull-Out
Math)
Limit Rigor - Focus
on Foundational
Skills (iReady/Go
Math)

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Note. Table 13 identifies specific data from each interview that aligns with the overall
system or methodology related to student learning.
Data Analysis and Findings of the Research Questions
The semi-structured interview questions developed by the researcher were derived
from the results of the literature review process outlined in Chapter II. There were
twenty-two (22) total qualitative questions presented, and these questions were
categorized into five (5) main sections. These sections included Special Education
Students, Economically Disadvantaged Students, External Factors, School Services and
Resources, and General questions for school and participant characteristics (Appendix F).
Responses to the questions in these categories led to the creation of the foundational
categories of school systems and methodologies presented in this chapter.
Research Question 1
The first research question of this qualitative analysis was “What instructional
strategies and methods do high performing schools employ to achieve high proficiency
rates in mathematics among all students?” This question was addressed generally by
reviewing the responses to all questions presented to the participants during the interview
process; however, for the purpose of this data review, responses provided for questions
from the External Factors, School Services and Resources, and General categories were
closely analyzed.
From the data provided, the researcher determined that there were three main
themes among all schools represented in the study that school leaders greatly attributed to
overall student success on 5th-grade standardized mathematics assessments. These

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themes were curriculum, assessment, and systematic extension and intervention
strategies.
It was determined by the researcher that high student achievement on
standardized math tests is significantly influenced by the implementation of a guaranteed
and viable curriculum that is both vertically and horizontally aligned across school
entities. A vertically aligned curriculum ensures that the knowledge and skills taught in
one grade build seamlessly into the next, promoting a coherent and cumulative learning
experience. Horizontal alignment ensures consistency across different classes and
teachers within the same grade level, providing all students with equal opportunities to
master the required content. This systematic approach reduces gaps in learning and
ensures that students are well-prepared for the material assessed in standardized tests,
ultimately leading to higher achievement. All three schools analyzed in this research
project described the importance of curriculum, and how it needs to be reviewed and
structured from a K-12 perspective.
It was also determined that the use of a local system of assessments allows
teachers to regularly obtain detailed data on student performance, which they can then
use to inform and adjust their instruction. These formative assessments provide real-time
feedback on what students have learned and identify areas where they may be struggling.
By analyzing this data, teachers can tailor their teaching strategies to address the specific
needs of their students, offering targeted support and intervention where necessary. This
responsive approach ensures that instructional practices are closely aligned with students'
learning needs, leading to more effective teaching and better student outcomes on
standardized math tests. Furthermore, assessment data can be used to adjust curriculum,

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group students by name and need, and identify specific standards and skills that the
majority of students struggle to master. In each school represented in this study, local
assessments were utilized to obtain data throughout the school year to better prepare
students well before the formal state standardized testing date. Assessments referenced
were teacher-created benchmarks to obtain data on student retention over time, diagnostic
tests to obtain data on student growth on math skills and content, and common formative
assessments used to obtain data on specific units or lessons taught to determine each
student’s level of proficiency on the skills assessed.
A systematic process for extension and intervention was the final strategy
determined by the researcher to be vital in ensuring high levels of mathematical
achievement. This strategy typically involves using reliable data to identify struggling
students early, then using that data to provide them with additional support through small
group instruction, tutoring, or other targeted interventions, and continuously monitoring
their progress. Interventions are designed to be flexible and responsive, adjusting to the
evolving needs of students as they work to master challenging concepts. By providing
timely and appropriate support, the schools identified in this project guaranteed that all
students had the opportunity to succeed, thereby improving their overall achievement on
standardized math tests. Structured support systems help close achievement gaps and
ensure that students do not fall behind, thus contributing to higher levels of academic
performance.
Research Question 2
The second research question of this qualitative analysis was “What instructional
strategies and methods do high-performing schools employ to achieve high proficiency

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rates in mathematics among students who are economically disadvantaged?” This
question was addressed by reviewing the responses to all questions presented to the
participants during the interview process; however, for the purpose of this data review,
responses provided for the four (4) questions from the Economically Disadvantaged
category, as well as the five (5) questions from the School Services/Resources category
were closely analyzed.
From the data provided, the researcher determined that there were three main
themes among all schools represented in the study that school leaders greatly attributed to
the success of economically disadvantaged students on 5th-grade standardized
mathematics assessments. These themes were curriculum, parent resources, and
intervention strategies.
Throughout the interview process, it became glaringly apparent that equal access
to a strong curriculum is fundamental in leveling the playing field for economically
disadvantaged students. When all students, regardless of their socioeconomic status, have
access to high-quality instruction and educational materials, it is much more likely that
they can develop the skills and knowledge necessary for academic success. A curriculum
that is both aligned to state standards and challenging for students not only provides the
necessary skills for mathematical development, but also fosters critical thinking,
creativity, and problem-solving abilities. This equity in educational resources helps
bridge the achievement gap, enabling disadvantaged students to compete on an equal
platform with their more affluent peers. Each school leader in this project described a
culture in which all students were held to a high standard. Regardless of status,

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disability, or previous success, students have the opportunity to take courses that are both
relevant to their interests and challenging to their abilities.
Family engagement practices and parent training opportunities play an important
role in supporting economically disadvantaged students in the three schools identified in
this research project. When schools actively involve parents in their children's education
and offer training on how to support learning at home, it creates a collaborative
environment that boosts student achievement. Educated and engaged parents are better
equipped to help with homework, advocate for their children’s needs, and reinforce the
importance of education. This partnership between home and school is especially critical
for disadvantaged students, as it can provide the additional support and motivation they
need to succeed academically and socially. A word that was frequently used when
addressing school/community relations was trust. Each of the three school leaders
recognized the value of a strong, working partnership between parents and the school
itself.
Early intervention programs provided by the schools and districts represented in
this study proved essential for addressing the unique challenges faced by economically
disadvantaged students. These programs, which include preschool education, tutoring,
and specialized services, aim to identify and address learning and developmental delays
as early as possible. Early intervention ensures that students receive the support they need
before falling too far behind, improving their chances of long-term academic success. By
investing in these programs, districts can mitigate the impacts of economic disadvantage
and help all students reach their full potential, leading to a more equitable and just
educational system.

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All participating schools in this study shared formal intervention programming for
families outside of the school day. In most cases, this was directly tied to Title 1 funding
and the requirement for school/family engagement; however, School C, which is the
school in this study with the largest percentage of economically disadvantaged students,
shared an early intervention structure that was very robust and fully accessible. All
district families have free access to two different early intervention programs, inclusive
of transportation services to and from the school. School C’s zero to three (0-3) early
intervention program takes place throughout the school year once per month for students
that are three years of age and under. Also, any child within the district that is between
the ages of three (3) and five (5) can attend a two-hour program on Tuesdays and
Thursdays each week throughout the school year. Furthermore, after-school tutoring is
available four (4) out of five (5) days each week for students in grades K-5. All three
programs are fully funded by the district and are well-attended on a yearly basis.
Research Question 3
The third and final research question of this qualitative analysis was “What
instructional strategies and methods do high-performing schools employ to achieve high
proficiency rates in mathematics among students with learning disabilities?” This
question was also addressed by reviewing the responses to all questions presented to the
participants during the interview process; however, for the purpose of this data review,
responses provided for the four (4) questions from the Special Education category, as
well as the five (5) questions from the School Services/Resources category were closely
analyzed.

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From the data provided, the researcher determined that there were three main
themes among all schools represented in the study that school leaders greatly attributed to
the success of learning support students on 5th-grade standardized mathematics
assessments. These themes were instructional strategies, curriculum resources, and
intervention strategies.
Each participating school in this research project referenced a high level of
inclusion among students with learning disabilities into the general classroom setting.
This was also followed by the belief that small group instructional settings should
continue to follow the agreed upon curriculum that was developed by the school and
district. With this as a prerequisite, instructional strategies play a crucial role in fostering
high mathematical achievement among students with learning disabilities. By providing
tailored teaching methods that accommodate the diverse learning needs of all students
and not just learning support students, student engagement and learning can take place at
a much greater rate. These strategies often include differentiated instruction and/or
flexible grouping where teachers have the ability to modify content, processes, products,
and learning environments based on data and the individual learning profiles of their
students. Techniques such as visual aids, manipulatives, and interactive activities can be
utilized among targeted groups of students to help them grasp abstract mathematical
concepts more concretely, and explicit instruction, where teachers use clear, direct
teaching methods and provide step-by-step demonstrations, can significantly enhance
understanding and retention of mathematical principles among groups of students that
benefit from this level of support. By using specialized strategies, teachers can create a

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more inclusive classroom environment that supports the unique learning profiles of all
students regardless of their learning challenges.
Curriculum resources proved equally vital in supporting high mathematical
achievement among the participating schools in this project. The resources referenced
include adaptive software, classroom materials, and supplemental supplies designed to
align with students' learning abilities and styles. Adaptive software, such as Xtra Math,
iReady Math, and enVision Math, offered personalized learning experiences that adjust to
the student's pace and level of understanding, providing immediate feedback and practice
opportunities to assist in the process of academic growth. Specialized classroom materials
such as Go Math textbooks or teacher-generated resources that use adaptive language,
visual supports, and scaffolded problems can make complex mathematical concepts more
accessible. Additionally, integrating real-world applications and problem-solving
activities into the curriculum helps students with learning disabilities relate to and
understand mathematical content more effectively, thereby enhancing their engagement
and motivation.
Targeted intervention strategies also came to the forefront for school leaders when
addressing specific learning challenges and ensuring that students with learning
disabilities reached high levels of mathematical achievement. These interventions
included one-on-one instruction, small group instruction, and co-teaching learning
environments that address learning goals and provide specific supports for students in the
regular education classroom. The use of progress monitoring through multiple assessment
strategies were also apparent in all three settings and directly tied to targeted intervention
for all students. Whether through the use of frequent common formative assessments,

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quarterly benchmark assessments, or recurrent diagnostic testing, available data enables
educators to identify areas where students are struggling and adjust their teaching
methods accordingly. Each school also shared scheduled time within their schedules to
provide extension and/or intervention for students outside of the regular math setting. By
implementing these practices, teachers have the ability to identify struggling learners at
many different points throughout their instruction, and can intervene in a timely manner
to ensure that no students fall behind in meeting educational benchmarks throughout the
school year.
Summary
Through the identification of three schools in Pennsylvania that met specific
demographic and achievement criteria for mathematics, the researcher was able to
categorize similar responses and triangulate data to identify key components that were
attributable to high levels of student achievement. This method assisted the researcher in
developing a comprehensive understanding of the factors that contribute to mathematical
proficiency and provided the researcher with insight on potential strategies for
improvement as described in the data analysis and finding for each identified research
question.
Although there are many components that may contribute to the academic success
and overall achievement of student groups in mid-level mathematics, the results of this
research project show a substantial alignment between high proficiency rates on state
standardized math tests and three specific components within school systems.
A guaranteed and viable curriculum that is accessible to all students is essential
for achieving high levels of learning in mid-level mathematics because it ensures

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consistency and equity in educational opportunities. Such a curriculum is aligned with
educational standards and goals, providing a clear roadmap for what students need to
learn and achieve. It is designed to be achievable within the school year, allowing
teachers to teach all necessary material without overwhelming students. Moreover, by
being accessible, the curriculum accommodates diverse learning styles and needs,
ensuring that every student, regardless of background or ability, can engage with and
understand the content. This inclusivity fosters an environment where all students can
build a strong mathematical foundation, develop critical thinking skills, and progress
together, ultimately leading to higher overall achievement in mathematics.
Assessment programs that measure student growth, proficiency of essential skills,
and retention of material over time are fundamental for achieving high levels of learning
in mid-level mathematics. These programs are essential to obtain valuable data that can
help educators understand how well students are grasping key concepts and skills and
how they are progressing over time. By regularly assessing student performance, teachers
can identify areas where students are excelling and where they may need additional
support. This ongoing monitoring allows for timely interventions and instructional
adjustments, ensuring that learning gaps are addressed before they widen. Furthermore,
assessments that measure retention help ensure that students are not only learning the
skills outlined in the agreed-upon curriculum but also retaining these essential skills for
future use, which is vital for building a strong mathematical foundation. Ultimately,
effective assessment programs support a cycle of continuous improvement, enabling
students to achieve and sustain high levels of proficiency in mathematics.

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Finally, guaranteed time for targeted intervention during the school day that is
separate from the regular instruction and learning activities provided within the regular
math classroom is critical for helping students achieve high levels of learning in midlevel mathematics. This dedicated intervention time allows educators to provide
personalized support tailored to the specific needs of each student, addressing learning
gaps and reinforcing foundational skills without disrupting the flow of regular classroom
instruction. It ensures that students who require additional help receive focused,
individualized attention, which is often difficult to provide within the constraints of the
standard classroom environment. By having a distinct period for intervention, teachers
can employ specialized strategies and resources that cater to diverse learning needs,
fostering a more inclusive and effective learning experience. This structured support
helps students build confidence and competence in mathematics, ultimately leading to
higher achievement levels and better long-term retention of mathematical concepts.

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CHAPTER V
Conclusions and Recommendations
Mid-level mathematics is vital for the academic success of students, as outlined
by the Pennsylvania Department of Education (PDE) state standards for grades 3 through
8, consisting of numbers and operations, algebraic concepts, geometry, and data analysis
and probability. Mastery of these areas enhances critical thinking and problem-solving
skills, which are foundational for advanced studies and essential across disciplines. In
Pennsylvania, proficiency in mid-level mathematics is linked to overall academic
achievement, career and college readiness and is crucial for students pursuing various
STEM careers that are abundant in the economy; however, standardized math scores
consistently fall below state standards starting in 5th grade, with the most significant
decline occurring between the 4th and 5th-grade cohorts during the 2021-2022 school
year. This phenomenon is not only present in the Kiski Area School District, but it is an
overall trend across school districts in the Pennsylvania for the 2021-2022 standardized
testing period.
This project addressed three key questions about the school systems and
instructional practices that contribute to academic proficiency in 5th-grade mathematics.
The literature review offered an in-depth look at instructional methodologies and school
factors associated with high student achievement at the primary and middle school levels.
Data analysis and results provided a detailed view of current practices in three specific
districts in Pennsylvania that achieved high student proficiency on state standardized
math tests for the 2021-2022 school year. The criteria for these three schools included
having an economically disadvantaged population exceeding 20% and a proficiency rate

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of 60% or higher in 5th-grade mathematics, as measured by the Pennsylvania System of
School Assessments (PSSAs) for the 2021-2022 school year. The research findings in
this project linked the research questions, literature, and collected data.
This chapter will present the research conclusions, including the potential
applications of the findings and their perceived importance to educational leaders. It will
also discuss the research limitations and conclude with recommendations for future
studies.
Conclusions
The research study analyzed qualitative data gathered through semi-structured
interviews with educational leaders from three distinct schools in Pennsylvania. The
subjects in these interviews had a high degree of influence on the high level of
mathematical achievement of 5th grade students during the 2021-2022 school year
because of their leadership roles within their buildings. The data will highlight the
instructional practices, school systems, and academic programs most credited with
fostering student success in mid-level mathematics. Multiple tables will be used
throughout this chapter to display the emergent themes for each research question based
on their perceived degree of impact and alignment to existing literature.
Research Question 1
The first research question of this qualitative analysis was, “What instructional
strategies and methods do high-performing schools employ to achieve high proficiency
rates in mathematics among all students?” Table 14 highlights the themes that emerged in
the analysis of this question when looking at the foundational categories of school
systems and methodologies created by the researcher.

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Table 14
Themes Supporting High-Performance Among All Students
All Students
Research Question

Foundational Category

System/Methodology
Vertical Alignment

Curriculum

Horizontal Alignment
Curriculum Development/Approach
Assessment Strategy

Instruction/Assessment

Instructional Strategy
Lesson Planning Strategy

RQ1: What
instructional strategies
Professional Learning
and methods do highperforming schools
employ to achieve high
proficiency rates in
mathematics among all
Resources
students?

Professional Development
Collaborative Approach
Coaching Strategy
Human Resource
Textual Resource
Curriculum Resource
Parent Resource
Growth Strategy
Retention Strategy

Student Learning

Intervention Strategy
Special Education Strategy
Engagement Strategy

Note. Table 14 identifies the themes that emerged through the interview process specific
to research question 1. Foundational categories and system/methodology themes that
emerged are highlighted in yellow.
All three of the schools represented in this study were primary schools serving
students in grades 2-5. For the purpose of this study, 5th-grade achievement data was
specifically analyzed due to the notable discrepancy between 4th and 5th-grade

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proficiency scores on state-standardized math tests during the 2021-2022 school year.
Since 5th grade was the final grade in these schools, vertical alignment became a focal
point, allowing teachers to collaborate across grade levels within the same building.
Moreover, with the exception of School A, two of the three schools participating in this
study had two or fewer primary schools in their district structure. This setup enabled 5thgrade teachers to access and collaborate with their grade-level colleagues daily,
significantly enhancing their ability to align curriculum and instruction horizontally
within their respective schools.
School A, however, has devoted much time and resources to the development of a
vertically aligned curriculum over the past five (5) years. The principal of School A
explained that her district employed two instructional coaches over this time period who
were specifically delegated to review the mathematics curriculum in grades three through
eight (3-8) and assist teachers in creating and implementing a plan that was both aligned
to state standards and local expectations. Although school A does not utilize this model at
the current time, it was apparent that teachers of mathematics at the primary and middle
school levels take much pride in both the vertical and horizontal alignment of their
curriculum in the area of mathematics. The principal of School A attributed much of this
success to the two individuals serving as data and instructional coaches across early grade
levels.
A mathematics curriculum that is aligned to Pennsylvania state standards and
vertically integrated with previous and future grade-level curricula ensures a cohesive
and comprehensive educational experience for students. This alignment ensures that
students build upon their knowledge systematically, leveraging previous skills to support

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higher levels of learning and minimizing gaps and redundancies. Collaboration among
grade-level teams is vital for the curriculum’s effective implementation, as it fosters
consistency, shared best practices, and a unified approach to teaching. When
implemented with fidelity, such a curriculum assures that all students within the grade
level receive equitable and high-quality instruction, promoting fairness and equal
opportunity for academic success across diverse classrooms.
Robust local assessment strategies were also noted in all three schools represented
in this study. Assessment strategies in education are crucial for understanding student
progress and identifying areas for improvement. All three schools in this research project
referenced local assessment methods to measure student growth, retention of essential
mathematical skills, and proficiency with high-level, integrative questions.
The utilization of formative assessments, such as weekly quizzes and student selfassessments, to regularly monitor and track student progress and retention of key
mathematical concepts was a key commonality among the schools represented in this
study. Additionally, they incorporated performance-based tasks that required students to
apply multiple mathematical principles to real-world problems, comprehensively
evaluating their depth of knowledge and ability to integrate various skills. Furthermore,
these schools utilized program resources that provided diagnostic tests to assess academic
growth in mathematical reporting categories so that instruction could be tailored to meet
students' individual needs. Through these multifaceted assessment strategies, the schools
have been able to gain a detailed understanding of student achievement, ensuring that all
students receive the support necessary to excel in mid-level mathematics.

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The final theme that emerged related to research question one (1) was a schoolwide system to address struggling learners. One of the critical questions in the
collaborative process of student achievement is how we will address students who have
not learned (Dufour et al., 2010). In each interview conducted in this research project,
concrete systems were identified to address this question. The principal of School A
described a system of intervention that occurs organically within math classes throughout
the school year. Teachers administer quarterly benchmark exams created using the
Standards Aligned System (SAS) website to ensure that questions are aligned to state
standards for math and that they require a higher depth of knowledge for students to show
proficiency. Students in School A are then flexibly grouped for periods of time based on
results so that instruction can be differentiated according to their needs and current level
of proficiency on the essential skills that are a part of the curriculum scope. In addition to
this method of group intervention, teachers in School A have the ability to access
students during their elective periods to provide more targeted interventions on identified
mathematical deficiencies. This strategy has been implemented as both small-group
remediation and one-on-one intensive intervention.
The principal of School B described a similar process of identifying students in
need of support based on data obtained using a program called Xtra Math. This program
is a fact fluency assessment that can be administered at the beginning of each unit of
instruction or at the beginning of each lesson. Teams of grade-level math teachers review
results and determine which students are in need of additional support to reach a higher
level of proficiency on a specific skill or set of skills. Principal B described a flexible
building schedule that allows thirty (30) minutes per day for targeted math intervention,

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also utilizing elective class time as described in School A. Additionally, School B offers
after-school tutoring in mathematics one day per week to all interested students at no cost
to the family. Principal B described the intervention system that is in place as invaluable
to the success of struggling learners.
The intervention strategy implemented by School C was a bit different than the
other schools represented in this study; however, the core value of addressing struggling
learners remains a priority among the teachers and staff. Local formative assessments are
frequently administered by teachers within the regularly scheduled math classes in which
students are heterogeneously grouped in a full-inclusion model. Twenty to thirty (20-30)
percent of the schools special education population is scheduled into smaller groups for
math instruction; however, the curriculum taught is the same in terms of scope and
sequence within those identified math sections. The interventions provided based on the
results of formative assessments occur three times each week as a thirty-minute math
remediation period that is built into the master schedule for all students. The Title 1
coordinator of School C also described a program called “Go Math Spiral Review” that is
implemented by all math teachers with fidelity on a recurring schedule to address
previous skills taught and ensure retention of mathematical concepts throughout the
school year.
The three common themes described by the leaders of each school represented in
this project relative to math achievement are:
•

The vertical and horizontal alignment of curriculum

•

The utilization of effective formative assessment strategies

MID-LEVEL MATHEMATICS ACHIEVEMENT

•

90

The implementation of an agreed-upon system of targeted intervention for
struggling learners.

A guaranteed and viable curriculum ensures that math instruction is consistent and
coherent across different grades and classrooms, allowing for a seamless progression of
mathematical skills and concepts. Formative assessment strategies provide teachers with
real-time feedback on student learning, enabling them to adjust instruction as needed to
reach more students at their individual level of understanding. Additionally, these schools
implement intervention strategies to support students who struggle with math, offering
personalized assistance and resources to help them catch up and succeed. Together, these
practices create a strong framework for math education that promotes high achievement
and continuous improvement among all students.
Research Question 2
The second research question of this qualitative analysis was, “What instructional
strategies and methods do high-performing schools employ to achieve high proficiency
rates in mathematics among students who are economically disadvantaged?” Table 15
highlights the themes that emerged in the analysis of this question when looking at the
foundational categories of school systems and methodologies created by the researcher.

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91

Table 15
Themes Supporting High-Performance Among Economically Disadvantaged Students

Research Question

RQ2: What
instructional
strategies and
methods do highperforming schools
employ to achieve
high proficiency rates
in mathematics
among students who
are economically
disadvantaged?

Economically Disadvantaged Students
Foundational Category
System/Methodology
Vertical Alignment
Curriculum
Horizontal Alignment
Curriculum Development/Approach
Assessment Strategy
Instruction/Assessment
Instructional Strategy
Lesson Planning Strategy
Professional Development
Professional Learning
Collaborative Approach
Coaching Strategy
Human Resource
Textual Resource
Resources
Curriculum Resource
Parent Resource
Growth Strategy
Retention Strategy
Student Learning
Intervention Strategy
Special Education Strategy
Engagement Strategy

Note. Table 15 identifies the themes that emerged through the interview process
specific to research question 2. Foundational categories and system/methodology themes
that emerged are highlighted in yellow.
Addressing students of low socio-economic status produced a different set of
themes than that of the other student groups analyzed in this project. When the questions
related to economically disadvantaged students were addressed, it became clear that
engaging this sub-group of students was of the utmost importance for sustained growth
and achievement.

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All three schools in this study have made concerted efforts to improve the
learning outcomes of economically disadvantaged students by providing access to a highlevel curriculum, engaging parents in the educational process, and ensuring free
accessibility to resources outside of school hours. By ensuring that all students, regardless
of their economic background, have access to rigorous and challenging coursework, these
schools are intentional in their efforts to bridge the achievement gap. They also foster
strong parental involvement through regular family events and frequent communication,
creating a supportive community where parents are active participants in their children's
education. Additionally, each school offers comprehensive after-school tutoring programs
that are free for all students, providing extra academic support and reinforcing classroom
learning.
A standout feature across these schools is their dedication to engaging parents in
the school process. They organize various family-oriented events and maintain consistent
communication with parents to keep them informed and involved. This approach helps
build a cohesive support network, ensuring that parents feel empowered and connected to
their children's educational journey. The schools also make educational resources
accessible beyond regular school hours, further supporting students' academic growth and
development.
Particularly notable is School C's strong commitment to early intervention
strategies. There is a distinct correlation between the timing of poverty and how this
impacts educational success and school completion. Early intervention can strongly
impact students' future success (Brooks-Gunn and Duncan, 1997). School C epitomizes
this concept by offering a robust early intervention program that provides free services to

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children within the school district starting at a very early age. Specifically, children ages
0-3 have the ability to attend educational programming at the school once per month, and
children ages 3 and older benefit from bi-weekly sessions lasting two hours each. This
early intervention is substantial in laying a solid foundation for future learning and
addressing educational needs from a young age. School C also includes free
transportation for families in the community, ensuring that all students can participate
regardless of their financial situation. In addition to these early interventions, School C
offers free after-school tutoring for all students, demonstrating a comprehensive approach
to supporting economically disadvantaged students both during the school day and
outside of school hours.
The results associated with improving the learning outcomes of economically
disadvantaged students centered on the theme of enhancing student accessibility to
various school services. Success in this area hinges on:
•

Access to a high-level curriculum

•

Engaging parents through family events and frequent communication

•

Offering free resources outside of the school setting.

Schools must ensure that all students benefit from rigorous coursework and foster strong
parental involvement to create a supportive community. These efforts collectively help
bridge the achievement gap and promote educational success for economically
disadvantaged students.
Research Question 3
The third and final research question of this qualitative analysis was, “What
instructional strategies and methods do high-performing schools employ to achieve high

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proficiency rates in mathematics among students with learning disabilities?” Table 16
highlights the themes that emerged in the analysis of this question when looking at the
foundational categories of school systems and methodologies created by the researcher.
Table 16
Themes Supporting High-Performance Among Students with Disabilities
Students with Learning Disabilities
Research Question

Foundational Category

System/Methodology
Vertical Alignment

Curriculum

Horizontal Alignment
Curriculum Development/Approach
Assessment Strategy

RQ3: What
instructional
strategies and
methods do highperforming schools
employ to achieve
high proficiency
rates in
mathematics
among students
with learning
disabilities?

Instruction/Assessment

Instructional Strategy
Lesson Planning Strategy
Professional Development

Professional Learning

Collaborative Approach
Coaching Strategy
Human Resource

Resources

Textual Resource
Curriculum Resource
Parent Resource
Growth Strategy
Retention Strategy

Student Learning

Intervention Strategy
Special Education Strategy
Engagement Strategy

Note. Table 16 identifies the themes that emerged through the interview process
specific to research question 3. Foundational categories and system/methodology themes
that emerged are highlighted in yellow.

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Although similar themes developed when addressing this question as compared to
research question 1, the mathematical achievement and success of students with learning
disabilities came with a higher focus on instructional and intervention strategies in this
scenario. Similar to students of low socio-economic status, access to the highest level of
education through a guaranteed and viable curriculum also became a recurring theme
throughout the research process.
Nolet and McLaughlin (2005) described the importance of the individualized
education (IEP) team in writing student goals, evaluating present levels of proficiency,
and providing intervention strategies to address learning gaps throughout a child’s
education. If these items can be addressed appropriately, students with disabilities can
benefit significantly from inclusion in the general classroom setting.
All three schools in this research project shared a commitment to providing
students with learning disabilities the opportunity to engage in a rigorous and relevant
curriculum that aligns with grade-level math standards. Each school referenced full
inclusion within the mathematics classroom, implementing strategies to deliver targeted
interventions based on data and proficiency assessments of essential skills. Although the
leader at School C described the utilization of small group math instruction for twenty to
thirty percent (20-30%) of the special education population, she was adamant that the
mathematics curriculum was not altered in these classrooms, and students had the same
exposure to a high level of rigor and relevance in these classrooms. A student with a
specific disability in math or reading at any of the three schools studied does not result in
a diluted curriculum or a slower-paced experience, which could lead to learning gaps
over time. Instead, all students are exposed to high-level questioning and practical

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mathematical applications, with embedded intervention processes to address learning
difficulties and remediate essential math skills.
When discussing the benefit of external resources to aid in curriculum delivery
and instructional practices, both School B and School C placed a significant emphasis on
additional programs tailored to support learning at each student’s level of mastery. Both
schools employ comprehensive technology programs that use diagnostic testing to assess
students' competency in specific math concepts, offering engaging practice and activities
based on their proficiency levels. School B uses the online program Xtra Math, whereas
School C utilizes a math resource called iReady Math. These programs can be used
independently by students and families or facilitated directly by teachers in classroom or
remedial settings. While School A did not detail specific programming for this purpose,
the principal emphasized that data from the school's benchmark testing allows teachers to
provide similar remediation through collaboration and collective planning.
As discussed in research question 1, all three schools demonstrated the
importance of using data to identify proficiency levels in math skills and concepts for
students with learning disabilities. Frequent formative assessments, summative
assessments, and benchmark assessments were important in determining which students
needed remediation or intervention in essential learning skills. Concrete examples of this
support include School A's flexible grouping strategy, School B's daily 30-minute math
intervention sessions, and School C's three weekly 30-minute targeted math remediation
sessions. These strategies underscore the importance of providing support to help
students stay on track for high levels of mathematical achievement.

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Based on the findings related to research question three (3), in order to foster high
rates of mathematical achievement among special education subgroups, schools must
provide these students with:
•

Access to a guaranteed and viable curriculum at their grade level

•

Frequent assessments in which the data is used to tailor instruction and provide
targeted intervention

•

Meaningful resources that assist in the learning process and promote growth
among individual students

When students are provided the necessary resources and supports, learning can occur at
high levels regardless of the presence of learning disabilities.
Limitations
There were several limitations that could have a significant impact on the overall
results of this project. The first limitation of this comprehensive research study is the
small sample size, as only three schools were analyzed. While these schools met the
criteria of having a 60% or higher proficiency rate on state standardized math tests and an
economically disadvantaged subgroup of at least 20% for the 2021-2022 school year, the
limited number of schools restricts the generalizability of the findings. A larger sample
size would have provided a more detailed data set, allowing for a more comprehensive
analysis and increasing the reliability of the study's conclusions. The small sample also
limits the ability to capture the variability and nuances across different schools, which
could affect the overall interpretation of the effectiveness of educational systems and
strategies.

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Another limitation pertains to the researcher’s interactions with school
administrators. Ideally, the researcher aimed to speak directly with the principals of each
identified school to gain insights into their leadership and instructional strategies;
however, the principal of School C was on medical leave, necessitating an interview with
the school's Title I coordinator instead. This substitution might have resulted in a
different perspective, potentially lacking the depth of knowledge or strategic vision a
principal might offer. Additionally, the principal of School B was new to his position
despite having prior involvement with the school at the district level. This recent
transition could mean that the principal was still in the process of implementing or
adapting strategies, possibly affecting the consistency and depth of information gathered
regarding the school’s performance and approaches. A potential approach that may have
avoided this limitation would be to include small teams of school leaders rather than
depending on one building administrator in the formal interview process.
A third limitation is that the percentage of student subgroups for each school in
the study varied significantly. Variations in the demographics and sizes of these
subgroups can influence the comparability of the schools and the study’s overall findings.
For instance, one school might have a higher percentage of economically disadvantaged
students compared to another, affecting resource allocation, teaching methods, and
student outcomes. These demographic differences can confound the results, making it
challenging to attribute differences in proficiency rates solely to the educational strategies
and school systems implemented. A more uniform distribution of subgroup percentages
would have enabled a more controlled comparison and clearer insights into the factors
contributing to academic success.

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Finally, this research study relied heavily on the results of the Pennsylvania
System of School Assessments (PSSAs) for mathematics proficiency, introducing another
limitation. Many other quantitative data sources exist such as local formative
assessments, student course grades, benchmark assessments, and content diagnostic tests;
however, the researcher chose to focus the quantitative portion of this study to
standardized test results because this data transcended the largest number of schools in
the state of Pennsylvania. It is also notable that the effectiveness of schools and districts
is primarily judged based on the performance of those entities and organizations based on
their performance on state standardized assessments.
Recommendations for Future Research
The results of this study provide much insight into the methods and practices that
can lead to higher levels of student achievement in mid-level mathematics, particularly
among economically disadvantaged students and students with specific learning
disabilities; however, there are three recommendations for future research that may
provide a more comprehensive, in-depth analysis of this phenomenon.
The first recommendation is to broaden the scope beyond a single snapshot view
of mathematics proficiency rates. Instead of relying solely on data from the 2021-2022
school year, researchers could incorporate a longitudinal approach, considering multiple
years of data to identify schools with consistent historical success in mathematics
proficiency. This longitudinal analysis would provide a more comprehensive
understanding of schools' performance trends over time, allowing for the identification of
factors contributing to sustained academic achievement. By examining trends across

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multiple years, researchers can better assess the stability and effectiveness of educational
practices within high-achieving schools.
Secondly, future research could delve deeper into the specific implementation of
enrichment and remediation practices within high-achieving schools. Rather than
focusing solely on proficiency rates, researchers could investigate the strategies and
interventions used by schools to support student learning and achievement. It became
glaringly clear in this research project that effective intervention strategies were
perceived as crucial in overall student achievement. By examining the specific practices
identified by each school, researchers can gain insights into the effectiveness of different
instructional approaches and interventions. This detailed examination can inform best
practices for promoting mathematics proficiency and address any gaps or inconsistencies
in current educational strategies.
Finally, by exploring student cohorts within a specific school system over two to
three years, growth and achievement in mathematics can be tracked across multiple
grades, and strategies and practices can be monitored more systematically. Researchers
can identify critical factors influencing academic outcomes by analyzing variables that
contribute to students' overall success or lack thereof during this time period. This
longitudinal analysis would provide valuable insights into the effectiveness of
instructional practices, curriculum alignment, teacher effectiveness, and other schoollevel factors in promoting student learning and achievement in mid-level mathematics.
By focusing on specific cohorts, researchers can also account for individual differences
and better understand how student characteristics interact with educational practices to
influence academic outcomes over time.

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Summary
The purpose of this mixed method research study was to identify three highachieving schools based on their performance in 5th grade on state-standardized math
assessments. The study aimed to uncover the instructional practices and systems that
contributed to these schools' success, particularly focusing on the general student
population as well as special education and economically disadvantaged subgroups.
Through an in-depth analysis, the researchers selected schools with an economically
disadvantaged subgroup representing at least 20% of their student population and a
proficiency rate of at least 60% or above on the 5th-grade standardized math assessment
for the 2021-2022 school year.
Three specific themes emerged from this study that were considered critical by
each participant in the overall success of their high-achieving schools. First, each school
implemented a guaranteed and viable curriculum aligned with state standards for
mathematics. This curriculum was consistent horizontally across different classrooms and
vertically across grade levels, ensuring that all students, regardless of their backgrounds,
had access to high-quality mathematical instruction. The alignment facilitated a seamless
educational experience and minimized gaps in learning progression from one grade to the
next.
Second, the schools had established robust assessment systems that enabled
teachers and staff to pinpoint learning deficiencies early and often. These systems
provided critical data that informed the development and application of targeted
intervention strategies. By addressing learning gaps promptly, each school was able to

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prepare students effectively for subsequent levels of instruction, fostering continuous
academic growth.
Lastly, the study highlighted the importance of engaging students and families
within the community. Schools created processes to ensure that families were aware of
available resources and understood the strategies being implemented to support their
children's academic success, specifically in the area of math education. This community
involvement proved essential in reinforcing educational efforts and promoting a
supportive learning environment for all students within their schools as well as in their
home environments.
Based on the results of this study, and as an administrative leader in the Kiski
Area School District, the researcher will devote more time and effort to the development
of community engagement strategies as well as the administration and data analysis of
local assessment approaches. The data provided in this project strongly supports that
improvement in these areas can lead to higher levels of math achievement among
economically disadvantaged students and students with learning disabilities. By
implementing the data-supported strategies outlined in this study, the Kiski Area School
District can assist students within identified subgroups and improve overall mathematics
achievement at a pivotal time in the developmental process.

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APPENDICES

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Appendix A
IRB Approval

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Appendix B
Participant Recruitment Email

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Appendix C
Principal A Consent to Participate in the Research Study

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Omitted for Confidentiality

Omitted for Confidentiality

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Appendix D
Principal B Consent to Participate in the Research Study

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Omitted for Confidentiality

Omitted for Confidentiality

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Appendix E
School Leader C Consent to Participate in the Research Study

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Omitted for Confidentiality

Omitted for Confidentiality

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Appendix F
Structured Interview Questions

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