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The intersection of curriculum, teacher, and instruction and its implications for student performance

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The intersection of curriculum, teacher, and instruction and its implications for student performance

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THE INTERSECTION OF CURRICULUM, TEACHER, AND INSTRUCTION AND ITS
IMPLICATIONS FOR STUDENT PERFORMANCE
by
Christin Hwang Colton

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

May 2022

Copyright 2022 Christin Hwang Colton

ACKNOWLEDGMENTS
Throughout the writing of this dissertation I have received a great deal of support and assistance.
I would first like to thank my chair, Dr. Sandra Kaplan, for guidance. My dissertation committee,
professors, and peers have been critical in my growth throughout this process. In addition, I
would like to acknowledge my colleagues who have allowed me to balance work and school.
Finally, I would like to offer my words of appreciation to my family, who have encouraged me
not only to start, but finish, this program while supporting me throughout.

iii
TABLE OF CONTENTS
Acknowledgments..................................................................................................................... ii
List of Tables .............................................................................................................................v

List of Figures .......................................................................................................................... vi

Abstract ................................................................................................................................... vii

Chapter I: Overview of the Study ..............................................................................................1
Background of the Problem ...............................................................................1
Statement of the Problem ...................................................................................5
Purpose of the Study ..........................................................................................6
Significance of the Study ...................................................................................7
Limitations and Delimitations............................................................................8
Definition of Terms............................................................................................8
Organization of the Study ................................................................................10

Chapter II: Review of the Literature ........................................................................................11
Curriculum .......................................................................................................11
Theorists ...............................................................................................13
Philosophies .........................................................................................15
Theories of Human Development ........................................................15
Learning Theories ................................................................................16
Common Core Standards .....................................................................18
Teacher Perceptions .........................................................................................18
Self-Efficacy ....................................................................................................19
Teacher Knowledge .........................................................................................21

Chapter III: Methodology ........................................................................................................24
Sample and Population ....................................................................................24
Instrumentation ....................................................................................27
Interview Guide ...............................................................................................28
Data Collection ................................................................................................31
Data Analysis ...................................................................................................31
Summary ..........................................................................................................32

Chapter IV: Results ..................................................................................................................34
Research Questions ..........................................................................................34
Conceptual Framework ....................................................................................35
Participants .......................................................................................................38
Results of Research Question One ...................................................................38
Theme One: Essential Concepts ..........................................................39
Theme Two: Teachers’ Role ................................................................41
Theme Three: Teacher Development...................................................47

iv
Discussion of Research Question One .............................................................49
Results of Research Question Two ..................................................................50
Theme One: Barriers ............................................................................50
Theme Two: Content Knowledge ........................................................54
Theme Three: Ideology ........................................................................55
Theme Four: Ideal Math Instruction ....................................................57
Discussion of Research Question Two ............................................................60
Summary ..........................................................................................................61

Chapter V: Discussion .............................................................................................................62
Findings............................................................................................................62
Findings of Research Question 1 .....................................................................62
Finding 1: Concept Development ........................................................63
Finding 2: Application, Fluency, and Student Debrief ........................63
Finding 3: Supplementing the Curriculum ..........................................64
Findings of Research Question 2 .....................................................................64
Finding 1: Time Constraints ................................................................64
Finding 2: Teacher Knowledge Gaps ..................................................65
Finding 3: Teacher Self-Efficacy .........................................................65
Finding 4: Student Knowledge Gaps ...................................................66
Finding 5: Teacher Ideology ................................................................66
Limitations and Delimitations..........................................................................66
Implications for Practice ..................................................................................67
Future Research ...............................................................................................71
Conclusion .......................................................................................................73

References ................................................................................................................................ 76

v
LIST OF TABLES
Table 1. Curriculum Philosophies............................................................................................13

Table 2. Participant Overview .................................................................................................27

Table 3. Research Study Overview ..........................................................................................29

Table 4. Interview Questions ...................................................................................................30

Table 5. Findings Summary .....................................................................................................74

vi
LIST OF FIGURES
Figure 1. Conceptual Framework ............................................................................................38

vii
ABSTRACT
This study seeks to understand the role of curriculum in student learning by applying the
instructional core framework, which states that there are only three ways to improve student
learning at scale: raising the level of the content, increasing the skill and knowledge of teachers,
and increasing the level of students’ active learning. The purpose of this study is to examine the
curricular components of Eureka Math, a Common Core-aligned math curriculum, to understand
what and why components are emphasized, adapted, removed, or supplemented by teachers.
Using a semi-structured interview guide, the researcher chose a non-probability, criterion-based,
purposeful sampling of seven partipants, who were interviewed to gather qualitative data. The
data was transcribed, coded, then organized into themes. Analyses revealed teachers emphasized,
deemphasized, and supplemented parts of the curriculum due to environmental factors, teacher
knowledge, teacher self-efficacy, and teacher ideology. Findings from the study suggest teacher
knowledge gaps in content and pedagogy, teacher ideology, and teacher self-efficacy should be
addressed further in both pre-service and in-service teacher education to support student
learning. Further research on these topics is also recommended.

1
CHAPTER I: OVERVIEW OF THE STUDY
Although we are in the era of Common Core standards with an emphasis on college and
career readiness, student achievement in math has remained relatively stagnant based on NAEP
assessments (NAEP, 2015). According to PISA (2015), the United States ranks 38 out of 71
countries in math. In 2009, only 26% of 12th grade students met or exceeded proficiency in math
(NCES, 2010). In 2017, California’s SBAC math proficiency rate was 37% (CAASPP).
In addition to the low levels of math proficiency within the United States, there is a
significant achievement gap amongst subgroups. The fourth grade NAEP data displayed a 26
point achievement gap between White and Black students, and a 19 point gap between White and
Hispanic students (NCES, 2017). Not only are all students done a disservice in mathematics
instruction, but the data also suggests specific subgroups are further underserved.
There are many potential contributing factors to the current state of mathematics,
including the curriculum being used in the classroom as well as the instruction being provided.
This study examines the intersection between the curriculum, teacher, and instruction in the
context of elementary math education. It examines the design of a math curriculum for essential
curricular elements as defined by notable theorists. Furthermore, it aims to identify teacher
perception of the curriculum as well as the dominant reasons why and how teachers modify
specific curricular components of the curriculum. The study seeks to understand the instructional
decisions of the teacher based on the curriculum provided, which ultimately impacts student
learning. Based on these findings, recommendations on how to support teachers in pre-service or
in-service contexts to improve math instruction can be provided.
Background of the Problem
History

2
Over the last century, math instruction has swung back and forth on a pendulum
regarding content and pedagogy. In the early years of the 20
th
century, American educational
establishments consistently advocated a progressivist agenda, strongly in favor of child-centered
discovery learning and against the systemic practice and teacher-directed instruction (Klein,
2003). In the 1930s, educational journals, textbooks, administration, and teachers advocated for
progressivism, thus supporting a curriculum centered on the needs and interests of children, not
on subjects (Kilpatrick, 1941). The prevalent belief was that subjects should be taught based on
practical value or interest and that they should be integrated in elementary school rather than
treated as separate subjects (Kilpatrick, 1930).
In the 1940s, the discovery of army recruits lacking skills for basic bookkeeping
prompted criticism of the progressivist agenda within education for a lack of academic content
(Klein, 2003). With advances in science, engineering, atomic energy, and other technologies,
there was a heightened emphasis on math education, and progressive education was forced to
retreat in the 1950s.
During the 1950s and 1960s, math curricula emphasized coherent logical explanations of
math procedures. With the introduction of calculus, there was a strong emphasis on theory and
exotic topics through New Math with little attention to basic skills or their application. In 1957,
Sputnik called attention to low-quality math and science instruction. In response, the National
Defense Education Act increased the number of sciences, math, and foreign language classes.
Math instruction shifted once again from theoretical to practical application.
As New Math died out, during the late 1960s to 1970s, progressive education regained
momentum until 1983. The National Commission on Excellence in Education released A Nation
at Risk was released, which pointed to a variety of education issues, including a lack of attention

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to basic skills. Textbooks were quickly upgraded to include more rigorous content. The 1989
National Council of Teachers of Mathematics (NCTM) standards initiated a strong focus on
basic skills and high standards. Commercial math curriculum aligned to NCTM standards were
quickly created and distributed through textbooks and programs across schools. Influenced by
the period and historical events, the theories behind education in the 20th century changed
drastically. Ultimately, by the late 20th century, NCTM standards identified a set of content,
skills, and knowledge to be taught and learned in schools.
Curriculum Adoption
In California, by law, there is a state adoption of curriculum every eight years in English
language, arts, mathematics, science, history-social science, visual and performing arts, health
and world languages (CDE). The most recent adoption of the math curriculum aligned to the
Common Core state standards through kindergarten to grade eight was completed in 2014. The
adoption process included the social content review, public review and comment, and education
content review. Once the recommendations are made, the state board either accepts or denies
them. Districts use the report from the state board that lists the texts and materials. K–8 public
schools may adopt using the state funding to go through the evaluation process to determine
which material they shall adopt.
The state legislature sets aside funds to cover the costs of supplying instructional
material, including textbooks for every K–8 public school student, and these funds can only be
used to purchase textbooks and materials recommended in the adoption process. If districts
purchase books that are not on the state-approved adoption list, they must use other sources of
funding. Since textbooks are expensive, many districts adopt texts that are on the state-approved
list.

4
Before the new adoption, the California Department of Education (CDE), revised the
California Mathematics Curriculum Framework, which is the official state guide on what, when,
why, and how to teach mathematics. Publishers develop new textbooks based on the California
Mathematics Curriculum Framework. There is a process for the state-approved curriculum, and
districts have to choose strictly within those options while selecting their math curriculum.
Common Core Standards
New Common Core state standards enacted in 2009 focus on fewer topics but require
students to demonstrate a deeper understanding of math concepts. The novel standards bring a
completely new set of academic standards for math compared to those of NCTM that require
students to demonstrate a greater depth of understanding. This calls for students to apply their
knowledge of mathematical concepts to solve problems and explain their reasoning. The shift in
focus from computation to problem-solving strategies requires teachers to learn new concepts.
The new math standards aim to address two ongoing issues in US education—quality and
equality (Akkus, 2016). According to The International Mathematics and Science Study
(TIMSS), mathematics standards of the highest achieving nations have three common features:
rigor, focus, and coherence, and these features can be found in the Common Core math
standards.
Historically, there have been inequalities in the opportunity to learn challenging content
in classrooms across the United States (Akkus, 2016). When students are never exposed to a
topic, it is impossible for them to learn it. The Common Core Math Standards allows for wider
access to accurate educational content with a common set of standards that positively promotes
higher quality assessment and textbooks. If implemented effectively, the new standards coupled
with the new curriculum could potentially reduce the inequalities in the state in content

5
instruction. Standards and curriculum, however, are only a part of the puzzle that leads to
learning.
According to Akkus (2016), less than half of elementary teachers felt well-prepared for
teaching math compared to 60% of middle school and 70% of high school teachers. Elementary
teachers have less content knowledge and confidence in math compared to their secondary
counterparts. In providing instruction, it is important to adapt the material for the development of
children in such a way that it can address content and practice, but it is difficult to do so if the
teacher feels unprepared.
Collopy (2003) stated that teachers’ use of curriculum materials may be influenced by
their goals, interests, values, and expectations of the curriculum and may enact lessons in
strikingly different ways than how the curriculum developer or educational reformers intended.
Teachers approaching a curriculum must possess text competence, which refers to the ability to
use textbooks strategically (Reichenberg, 2016). Even though teachers receive high-quality
curriculum aligned to Common Core State Standards, they must understand the meaning and
interpret that curriculum to provide instructions to students. Due to a variety of variables, there is
a wide continuum of approaches in the instruction of the same curriculum with varied results,
depending on the teacher.
Statement of the Problem
There is a wide variety of mathematics curricula across different schools and districts,
and differences in the implementation of the same curriculum within the same school site.
According to California SBAC results, 37% of students scored proficient in math in California,
with only 20% proficient at Corner Street Charter School (CDE). In response to dismal results in
math proficiency, there has been an emphasis on the widespread adoption of new curriculum

6
material as a primary strategy for improving math (Remillard, 2005). Corner Street Charter
School adopted a new math curriculum, the Eureka Math, school-wide at the beginning of the
2017–2018 school year. The Eureka Math was chosen based on the alignment of focus,
coherence, and rigor in the standards, which are the three common features of the highest-
achieving nations according to the International Mathematics and Science Study (TIMSS). The
math curriculum is a critical component in student learning and must be examined for quality,
however, additional variables must also be considered when attempting to increase student
learning. The curriculum alone will not be sufficient to increase student learning.
Remillard (2005) stated that the new standards require teachers to play a substantially
different role in a mathematics classroom than ever before as the new curriculum emphasizes
mathematical thinking, reasoning, conceptual understanding, and problem-solving in realistic
contexts. Being the medium between curriculum and instruction, teachers make modifications to
the curriculum given the circumstances, their content knowledge, and their personal beliefs.
Remillard (2005) stated that it would be inaccurate and irresponsible to conclude that all the
interpretations of a written curriculum are equally valid.
Purpose of the Study
This dissertation examines the curricular components of the Eureka Math, a Common
Core-aligned math curriculum with an emphasis on focus, coherence, and rigor. It seeks to
understand the driving forces behind the instructional decisions of teachers as they make
meaning and interpret the curriculum to teach their students. In addition, it hopes to uncover the
critical elements of the curriculum that are removed, modified, supplemented, and delivered, and
to assess the potential impact of these changes on student learning within the context of a TK–
6th grade school in South Los Angeles. This will provide insight into the kind of support that

7
teachers require in both pre-service and in-service contexts to provide high-quality math
instruction that increases students learning and performance, while simultaneously decreasing
the achievement gap amongst certain subgroups.
Research Questions:
1. What components of Eureka Math do teachers emphasize or deemphasize?
How do teachers remove, adapt, or supplement certain components of Eureka
Math?
2. What are the dominant reasons behind teachers modifying certain curricular
components of Eureka Math for their class?
Significance of the Study
An underlying cause of the math achievement gap is unequal opportunities to learn
mathematics, specifically for low-income, Latinx and African American students (Flores, 2007).
Flores (2007) discussed the concept of the opportunity gap where African American and Latino
students have teachers who provide low-quality mathematics instruction using inappropriate
resources, with a little emphasis on reasoning and problem-solving. National Council of
Teachers of Mathematics (NCTM) vision of reform in mathematics is based upon constructivist
approaches in which teachers are a critical component (Swars, 2005).
This study will examine teachers’ decision-making processes as they alter, adapt, or
translate the curriculum offerings to make their instruction more “appropriate,” based on their
characteristics, knowledge, capacities, beliefs, perceptions, and experiences. Teacher efficacy is
also a factor in instructional practices implemented (Swars, 2005). Currently, there is limited
research regarding the interrelationship between the curriculum and teacher and its impact on
instruction. This study will give us a better understanding of teachers’ decision-making process

8
in modifying curriculum for instruction and its implications. This study would benefit teacher
educators and teachers themselves in understanding what misconceptions and needs teachers
have when modifying curriculum, which would ultimately benefit students.
Limitations and Delimitations

A limitation of this qualitative study is that I, the researcher, am the primary instrument
of the study. As the primary instrument, the researcher may mistake the meanings of words that
have multiple, ambiguous meanings, which can lead to misinterpretation of data. Although I
have specifically chosen to interview teachers who are not at my school site due to the hierarchy
of roles, I am acquainted with the teachers I shall be interviewing at the other school site, which
may influence their interview responses.
The criteria for the purposive, convenience sample is a TK–6th grade math teacher at
Corner Street Elementary School using Eureka Math. A delimitation of the study is the small
sample size and voluntary participation that does not allow for generalizability.
Definition of Terms
For a better understanding of this study, the following terms are operationally defined.
Common Core Standards refers to an educational initiative from 2010 that details what K–12
students throughout the United States should know in English language arts and mathematics
after each school grade.
Common Content Knowledge (CCK) refers to the mathematical knowledge and skill used in
settings other than teaching (Ball et al., 2008).
Critical Reconstructionist Speaker refers to students themselves and the concerns, interests, and
injustices they feel as the starting point for meaningful and worthwhile learning.

9
Curriculum refers to all of the educative experiences learners gain in an educational program, the
purpose of which is to achieve broad goals and related specific objectives that have been
developed within a framework of theory and research, past and present professional practice, and
the changing needs of society (Parkay et al., 2014).
Essentialism refers to learning of essential content comprised of basic information selected from
past. Learning takes place through learner experiences decided by teacher. Pupils are docile and
obedient, and interest grows out of efforts to learn (Bagley, 1941).
Experientialist refers to the natural way of learning and teaching—one that we all experience
outside of formal learning contexts (Schubert, 1996).
Intellectual Traditionalist refers to the acknowledgment of the power of the classics and the great
ideas embedded in them (Schubert, 1996).
Knowledge of Content and Students (KCS) refers to the combination of knowing about students
and knowing about mathematics (Ball et al., 2008).
Knowledge of Content and Teaching (KCT) refers to the combination of knowing about teaching
and knowing about mathematics (Ball et al., 2008).
NAEP refers to National Assessment of Educational Progress.
Progressivist refers to focusing learning on child’s interest, problems, and purpose and
supporting their ability to think independently and intellectually. It starts with student’s
interests, then assumes more socially responsible roles (Kilpatrick, 1941).
PISA refers to Programme for International Student Assessment.
Social Behaviorist refers to identifying what knowledge, skills, and values lead to success in
each generation (Schubert, 1996).

10
Subgroups refers to any group of students who share similar characteristics, such as gender
identification, racial or ethnic identification, socioeconomic status, physical or learning
disabilities, language abilities, or school-assigned classifications.
Specialized Content Knowledge (SCK) refers to the mathematical knowledge and skills unique to
teaching (Ball et al., 2008).
Organization of the Study
Chapter 1 of the study will provide background about the history and previous research
on curriculum and instruction within math. The history describes the changes in content and
pedagogy over the last century leading up to commercial math curriculum aligned to NCTM
standards, followed by the adoption of Common Core state standards. Then it will introduce the
statement of the problem, the purpose of the study, and the significance of studying the
intersection of curriculum, teacher, and instruction to address the nation’s stagnant performance
in math education as well as the achievement gap. Chapter 2 will provide the conceptual
framework for the research questions, which includes the intersection of the curriculum with
teacher ideology, teacher knowledge, and teacher self-efficacy. Chapter 3, on methodology, will
give an overview of the qualitative study being conducted. Teachers at Corner Street Elementary
School participated in a semi-structured interview to gather qualitative data. In Chapter 4 will
describe data collection and analysis through first level and second level coding that lifts out
themes connected to the research question. Nvivo was used to organize the data into notes and
themes. Chapter 5 will summarize the findings, implications, and topics for further research.

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CHAPTER II: REVIEW OF THE LITERATURE
To understand the curricular components of Eureka Math and how students learn, it is
important to review the literature on human development and learning theories. In addition,
curricular theorists and their curricular philosophies, including their beliefs about the learning
process and learning experience, have impacted approaches to curriculum and instruction in the
United states in the past century.
Various factors have shaped a wide range of curriculum and instruction, including
historical events and the beliefs of popular curricular theorists. Beliefs about the learning
process, learning experience, and assessing learning are implicit in the curriculum. The
background information is important in answering the research question about the essential
components of Eureka Math, how teachers modify certain curricular elements, and the dominant
reasons for these modifications. The findings of the present study connect to the literature
through their implications for student learning of math.
Curriculum
Donovan and Bransford (2005) stated that for the students to develop competence in an
area of inquiry, students must have a deep foundation of factual knowledge, understand facts and
ideas in the context of a conceptual framework, and organize knowledge in ways that facilitate
retrieval and application. A metacognitive approach to instruction can help students learn to take
control of their learning by defining learning goals and monitoring their progress in achieving
them (Donovan & Bransford, 2005). Teacher knowledge, ideology, and self-efficacy impact
instruction and its ability to be learner-centered, knowledge-centered, and assessment-centered as
recommended by Donovan and Bransford. Reference Table 1 for the different curriculum
philosophies and educational theorist, and how it aligns with the Eureka Math curriculum.

12
Table 1
Curriculum Philosophies
Progressivist Essentialist/
Traditionalist

Behaviorist Critical
Reconstructionist
Constructivist
Cognitive theory
Philosophy Emphasizes the
importance of
studying children
to identify their
interests,
problems, and
purpose

Children’s
interests must be
identified so they
can serve as the
focus of
educational
attention.

Data points to
measure learning,
including
observations,
interviews, and
questionnaires.

Learning takes
place through the
active behavior of
students.

Three criteria for
learning
experiences:
continuity,
sequence,
integration.
Basics learned
selected to from
what has been
valued in the past

Learning takes
place through the
experiences that
the learner has
which the teacher
decides to
provide.

The attitude of
pupils must be
docile, receptive,
obedient.
The curriculum
should teach the
knowledge
skills, behaviors
and values that
help students
become
successful in
current society.
Emphasizes the
concerns, interests,
injustices students
identify to be the
starting point for
meaningful and
worthwhile learning.
Enactive
representation
(action-based)

Iconic
representation
(image-based)

Symbolic
representation
(language-based)

Learning to learn
Theorists Tyler
Dewey
Bagley
Schubert
Jerome Bruner
Elements in
Eureka Math
Three criteria for
learning
experiences:
focus, coherence,
application
Yes No No Concrete,
Pictorial, Abstract

Student Debrief

13
Theorists
Tyler, a progressivist, emphasized the importance of focusing attention on studying
children individually to identify their interests, problems, and purpose (Tyler, 1949).
Contrastingly, an essentialist views objectives as the basic learning selected from the past with
the belief that learning takes place through the experiences of the learner and those the teacher
decides to provide. The Eureka Math curriculum does not take students’ interests into account;
rather, it identifies the essential content to be taught.
A learner’s experience is the interaction between the learner and the external conditions.
Learning takes place through the active behavior of the students that leads to learning. In order
for learning to occur, students must have experiences that allow them to practice the kind of
behavior implied by the objective, and an opportunity to deal with content. The learning
experience must also give student satisfaction as a result of carrying on the kind of behavior
implied by the objective.
Many specific experiences can be used to attain the same educational objective, and the
same learning experience can bring about several outcomes. Tyler (1949) listed the three criteria
for learning experiences as: continuity, the vertical reiteration of major curriculum elements;
sequence, successive experience builds upon the preceding one; and integration, application in
real-world contexts. In Eureka Math, these elements are termed focus, coherence, and
application.
Tyler (1949) also suggested using more than a single data point to see if learning, a
measured change in behavior, has taken place. These data points can include observations,
interviews, and questionnaires. Eureka Math includes daily assessments in the form of exit
tickets and end-of-module assessments, which are the more traditional approaches to gathering

14
student data. Tyler (1949) held that evaluation is a critical element of learning. He stated that it is
important to examine data to suggest possible explanations or hypotheses about the reason for a
particular pattern of strengths and weaknesses in the plan as well as to determine what was
effective and what needs improvement.
The curriculum encompasses the course of study, course content, planned learning
experiences, and intended learning outcomes (Parkay et al., 2014). Curriculum and instruction
work interdependently where the curriculum is the “what” and instruction is the “how.” The
curriculum is focused on social forces, theories of human development and theories of learning
and learning styles. Although most people would agree that one goal of the curriculum is to
prepare students for the figure, there has been little consensus about the kind of knowledge and
skills that will be required in the future.
Schubert (1996) stated that curriculum is beyond a textbook and concerns what is worth
knowing, experiencing, doing, and being. An intellectual traditionalist believes in the power of
the classics and the ideas embedded in them. The behaviorist would state that the curriculum
should teach the knowledge, skills, behaviors, and values that help students become successful in
current society while experientialist and critical reconstructionist emphasizes the concerns
interests, injustices students identify to be the starting point for meaningful and worthwhile
learning (Parkay et al., 2014).
Bagley (1941) held that interest grows out of efforts to learn. It is the adult’s
responsibility for guidance and direction because of the necessary dependence. Although self-
discipline is the ultimate goal, imposed discipline is a necessary means to this end. Kilpatrick
(1941) believed that learning is most effective if it addresses students, purpose, and concerns.
With that belief, he stated that curriculum begins with children’s natural interests, gradually

15
preparing them to assume more socially responsible roles and become worthy members of
society (Kilpatrick, 1941). In addition, he believed that the curriculum should teach students to
think intelligently and independently through opportunities to practice and planned jointly by
teachers and students (Kilpatrick, 1941).
Philosophies
Two conflicting educational philosophies present in the 20th century were traditional and
progressive. Traditional or essentialist education is rooted in the foundation that the subject
matter of education consists of bodies of information and skills that have been worked out in the
past; therefore, the chief business of the school is to transmit this knowledge to a new generation,
preparing the young for future responsibilities and success in life, while putting into practice the
organized bodies of information and prepared forms of skill that apply the material of
instruction. For this purpose, the attitude of pupils must be docile, receptive, and obedient.
Books, especially textbooks, are the wisdom of the past to be used. Teachers are merely the
agents through which knowledge and skills are communicated, and rules of conduct enforced to
students. Progressive education, on the other hand, stresses the importance of learning, beginning
at the child’s interests with opportunities for the child to engage in active learning.
Theories of Human Development
Maslow held that first and foremost the biological and psychological needs have to be
satisfied before a child can learn. The basic need for survival and safety comes first. Piaget’s
model of cognitive development states that children learn through interacting with their
environments; their thinking progresses through a sequence of four cognitive states (Wadsworth,
1971). Stage 3, concrete operations, occurs at ages 7–11, when the child explores and masters
concepts of objects, numbers, time, space, and causality and can use logic to solve problems .

16
This aligns with the concrete focus in Eureka math. Stage four occurs at ages 11–15, the formal
operational stage when children can make predictions, think hypothetically, and reason
abstractly, which aligns with the abstract components in Eureka Math. The Eureka Math
curriculum focuses on three components to build conceptual understanding: concrete, pictorial,
and abstract.
Learning Theories
Bandura (1986) advocated social learning theory, which focuses on modeling or
observational learning, where students observe then imitate their teachers. Mental modeling
provided by the teacher includes demonstrating to students the thinking involved in a task,
making students aware of the thinking involved, and focusing students on applying their
thinking.
Bruner (1961), a perennialist, discussed the concept of intellectual potency which is about
teaching kids to learn how to learn, which requires the active involvement of the learner and the
ability to connect to oneself. This allows children to transfer their knowledge to life. Students
learn through inquiry in context, and the degree of involvement equates to the degree of
knowledge gained (Bruner, 1961). Bruner subscribed to cognitive learning theory, which focuses
on the unobservable processing, storage, and retrieval of information from the brain. It
emphasizes personal meaning, discovery, “advanced organizers,” and generalizations. Bruner
valued learning activities of high intellectual quality and teaching for understanding. Authentic
learning tasks enable students to see the connections between classroom learning and the world
beyond the classroom, both now and in the future.
Donovan and Bransford (2005) stated that students come to the classroom with
preconceptions about how the world works. If their initial understanding is not engaged, they

17
may fail to grasp the new concepts and information, or they may learn them for purposes of a test
but revert to their preconceptions outside the classroom. To develop competence in an area of
inquiry, students must have a deep foundation of factual knowledge, understand facts and ideas
in the context of a conceptual framework, and organize knowledge in ways that facilitate
retrieval and application. This metacognitive approach to instruction can help students learn to
take control of their learning by defining learning goals and monitoring their progress in
achieving them. To summarize, Donovan and Bransford recommended a learner-centered,
knowledge-centered, and assessment-centered instruction that engages prior understanding,
builds factual knowledge and conceptual frameworks in understanding, and allows for students
to self-monitor their learning.
Tyler (1949) discussed the four critical questions that must be answered when creating a
curriculum: What educational purposes should be attained or what is the objective? What type of
learning experiences accomplish these purposes or objectives? How should the learning
experiences be organized? And how will the learning experience, purposes, objectives be
evaluated? These questions must be at the core of curriculum planning so it includes the what
(purpose and objective) and the how (organized learning experiences). From there, the evaluation
determines whether the goal was met and to what degree.
In addition, the learners’ culture, basic achievement level, and interests are all important
factors to consider when planning a curriculum. Dewey (1902), a progressivist, believed that the
starting point of education was a child’s own experiences, and the primary equipment of
education was a child’s interest. He believed in a child- and community-centered curriculum.
Greene (1971) described education as making sense and meaning of the world and working
towards a transformation of the world with new perceptions.

18
Common Core Standards
The draft of the new Common Core state standards released in 2010 focused on fewer
topics, but require students to demonstrate a deeper understanding of math concepts (Parkay et
al., 2014). The Common Core State Standards Initiative (CCSSI) stipulates that the standards:
are aligned with college and work expectations; are clear, understandable, and consistent; include
rigorous content and application of knowledge through higher-order skills; build upon strengths
of and lessons learned from current state standards; are informed by other top-performing
countries, so that all students are prepared to succeed in a global economy and society; and are
evidence- and research-based.
Teacher Perceptions
Race is significant in the context of curriculum as there are great inequities that exist
between the schooling experiences of White middle-class students and those of poor African
American and Latino students (Ladson-Billings, 1999; Ladson-Billings & Tate, 2006). Some
districts in California have a large student population from an ethnic or cultural background that
differs from their teachers, who are majority White (Flores, 2007). Teachers assume deficits in
the students, to which they attribute the failure of a student to thrive intellectually, rather than
locating and teaching to their strengths, such as resilience, eagerness, energy, and creativity
(Flores, 2007). Franke et. al (2015) reported that urban teachers bring deficit views about
students of color with them that need to be challenged at the same time as they transform content
instruction. Math instruction for low socioeconomic status (SES) students of color emphasizes
disconnected concepts, math vocabulary out of context, following steps, and answers over
explanations (Franke et. al, 2015). Teachers working with students of color who are not
succeeding often collect evidence that leads them to place the blame on students and their

19
families without considering structural forces that shape success. Specifically in urban schools
serving low SES students of color, teacher ideology and its impact on instruction must be
examined.
Remillard (2005) suggested that there is a tendency for school districts to regulate
mathematics teaching practices by mandating the use of a single curriculum in response to a
failure of schools to raise achievement levels, particularly for students of color and in low-
income communities. There is an emphasis on widespread adoption of new curriculum material
as a primary strategy for improving math, but such information and content are unfamiliar or
foreign to most teachers, who are an integral component of instruction.
Self-Efficacy
Research has found that teachers’ past experiences with mathematics and perceptions of
teaching effectiveness were associated with mathematics teacher self-efficacy (Swars, 2005).
Research suggests highly efficacious teachers are more effective mathematics teachers than
teachers with a lower sense of efficacy (Swars, 2005). Teachers with high levels of efficacy have
demonstrated different characteristics related to work ethic and pedagogical practice than
teachers with low levels of self-efficacy (Swackhamer et al., 2009).
Swars (2005) stated that teacher efficacy also has been correlated to instructional
strategies and willingness to embrace innovation. Highly efficacious teachers are more likely to
use inquiry and student-centered teaching strategies, while teachers with a low sense of efficacy
are more likely to use teacher-directed strategies (Swars, 2005). A teacher’s self-efficacy for
student engagement, classroom management, and instructional strategies is a significant
predictor of teachers’ willingness to implement curriculum reform (Cerit, 2013). Teachers with

20
high teaching efficacy are more likely to try new teaching strategies, particularly techniques that
may be difficult to implement and involve risks such as sharing control (Swars, 2005).
Different expectations for the different students are often reflected in the ways teachers
provide instruction. Teachers need to help students develop a relational understanding of
concepts, number sense, and an environment where students are able to justify their solutions and
question other students about their thinking (Flores, 2007). Mathematics instructional strategies
that provide authentic mathematics activities in real-world situations and the instructional
strategy of using manipulatives as an aid to teaching and learning in the mathematics classroom
are part of quality math instruction.
Howe (1999) suggested that Chinese teachers have a much better grasp of mathematics
they teach than do American teachers. For example, fewer than 20% of U.S teachers had a
conceptual grasp of the regrouping process, compared to 86% of Chinese teachers. Teachers’
mathematical knowledge is central to how well they can use curriculum materials, assess student
progress, and judge how to present, emphasize, and sequence the material. Preservice teachers
need positive experiences within mathematical methods courses to build efficacious towards
teaching mathematics (Swars, 2005). For in-service teachers with lower levels of specific content
knowledge, content courses that are designed to support a teachers’ development of content
knowledge and pedagogy can be a valuable way to increase levels of self-efficacy (Swackhamer
et al., 2009). Professional development or additional education that impacts a teachers’
understanding of their craft can affect the teacher’s perceived ability level and increase self-
efficacy (Swackhamer et al., 2009).
Slavin and Lake (2007) examined three math approaches and identified several methods
with strong positive outcomes on student achievement, which included cooperative learning,

21
classroom management, and motivation. In addition, there were positive outcomes in student
achievement programs that helped teachers understand math content and pedagogy (Slavin &
Lake, 2007). Although there were different approaches that had positive outcomes on student
achievement, Slavin and Lake (2007) identified several beneficial instructional process strategies
employed by teachers, such as using time effectively, keeping children productively engaged,
giving children opportunities and incentives to help each other learn, and motivating students to
be interested in learning mathematics.
Teacher Knowledge
Charalambous and Hill (2012) described teacher knowledge, curriculum, materials and
quality of instruction as a complex relationship that needs to be unpacked. Teacher knowledge
and curriculum materials individually and jointly contribute to instructional quality. Curriculum
alone does not guarantee high-quality teaching, but the dynamic interrelationship between the
teacher and curriculum shapes both; each is an active participant contributing to instructional
outcomes (Charalambous & Hill, 2012).
Franke (2015) suggested that promoting mathematical learning requires teachers to
engage in “productive struggle” through communication with others. Communication with others
requires explaining one’s thought process and discussing other students’ reasoning processes. US
Common Core Standards for Mathematical Practice call for students to “construct viable
arguments and critique reasoning of others,” requiring explanation and analysis, explaining ideas
to others, and considering one’s own ideas concerning others’ ideas.
Levels of engaging in this type of discourse have implications for student outcomes. It is
important to support students in engaging with each other’s mathematical ideas on a detailed

22
level, looking beyond the interactions of students themelves first move and towards how teachers
extend their interactions with students to support opportunities for productive struggles.
Remillard (2005) suggested that there is little research on how teachers use curricula or
the teacher curriculum relationship. In the 1950s, the curriculum was “teacher proofed” and seen
as fixed, the teacher being a conduit for curriculum, rather than a user or designer (Remillard,
2005). Historically, teachers have stuck to published curricular materials, and how teachers
interact with and use this curriculum has not always been explored because it has not always
been considered relevant to understanding.
Knowledge, attitudes, and perceptions of their pedagogical and social context weigh
heavily in the teachers’ reasoning. Teachers bring their own beliefs and experiences to their
encounters with the curriculum to create their meaning, frequently interpreting the intentions of
authors. This suggests a need for further studies that can examine what teachers do as they alter,
adapt, or translate textbook offerings to make them appropriate for their students. The process
includes selecting and designing tasks, then enacting the task, improvising in response to student
responses. There were few efforts from administrators and teachers devoted to examining and
conceptualizing curriculum materials. Individual teachers’ characteristics, including beliefs,
knowledge of math, and professional identity, impact curriculum use.
Teachers need a variety of knowledge types for effective math instruction (Grossman,
1990; Hill et al., 2004; Shulman, 1986). Ball, Thames, and Phelps (2008) stated that strong
knowledge of basic mathematical content does matter; however, additional capabilities must be
layered on top of that foundation, including mathematical knowledge for critical thinking and
application. Grossman (1990) observed that the four different knowledge types are common
content knowledge, specialized content knowledge, knowledge of content and students, and

23
knowledge of content and teaching. For teachers to effectively interpret the curriculum, they
must have all of these knowledge types.

24
CHAPTER III: METHODOLOGY
This study examines the characteristics of Eureka Math, a Common Core-aligned math
curriculum, for curricular components. By utilizing seven teacher interviews and document
analysis in a low-performing TK–6th-grade charter school, it seeks to understand how teachers
make meaning and interpret the curriculum and the dominant reasons how and why they modify
the curriculum, in relation to perceived barriers to learning, teacher knowledge, ideology, and
self-efficacy.
The qualitative method is research focused on discovery, insight, and understanding from
the perspectives of those being studied through an inductive process (Merriam & Tisdell, 2015).
To answer my research questions, which seek to understand the intersection of curriculum,
teacher, and instruction in the context of math, this is the appropriate approach. Qualitative
methods allow for the construction of meaning in the context of the study through inquiring or
investigating into something in a systematic manner (Merriam & Tisdell, 2015). Ultimately, the
purpose of this study is applied research: to potentially improve the quality of the practice of a
particular discipline (Merriam & Tisdell, 2015).
Sample and Population
A non-probability, criterion-based purposeful sampling was used for my study in that all
teachers were math teachers in a TK–6th urban setting (LeCompte & Schensul, 2010). The
convenience sample was selected based on time, location, and availability. The seven teachers
were selected through purposive and convenience sampling to discover, understand and gain
insight into the research question (Merriam & Tisdell, 2015). The teachers at various grade
levels, years of experience, and comfort levels in math were purposely selected as information-

25
rich sources. All teachers taught the same curriculum, which allows for comparison on how the
curriculum is interpreted and carried out.
All teachers work at Corner Street Charter Elementary, which is a TK–6 Public School
Choice 2.0 school in Los Angeles, part of a larger charter network. Corner Street Charter
Elementary has served the community of South Los Angeles since 2011. “Corner Street Charter
is a safe, caring, academically rigorous, and inspiring learning environment where students
develop the skills, knowledge, and traits to become college-educated leaders in their
communities. The school’s mission is to provide all students with an exceptional education that
will allow them to excel inside and outside the classroom. The school seeks to achieve this
mission by providing students with a rigorous core curriculum, a well-trained staff, high
standards and expectations, extended instructional hours and support, personalized learning
opportunities, and early access to college-preparatory experiences. By ensuring students become
voracious, self-motivated, competent and lifelong learners, the school will prepare them not only
for college but also for the 21st century world” (LCAP).
Corner Street Elementary currently serves 347 K–6th students, and the demographics are
as follows: 99% Free Reduced Lunch; 90% Latino; 10% African American; 56% English
Learners, and 9% SWD. Corner Street Elementary is currently implementing a variety of
evidence-based instructional resources that support student learning in dynamic ways. All
instructional materials for ELA, math, writing, technology and English language development
are very current and are explicitly aligned to the CCSS (LCAP). In 2017, 20% of students scored
proficient in mathematics.
The charter network is a non-profit organization that operates high performing charter
schools that focus on one goal: preparing urban students for college. The charter network’s

26
mission is to open and operate small, high-quality charter schools in low-income neighborhoods
to increase the academic performance of underserved students, develop effective educators, share
successful practices with other forward-thinking educators, and catalyze change in public
schools.
The first level of access required was the school principal. He was informed that a study
regarding curriculum and instruction was being conducted. A request was submitted to reach out
to the teachers at the site, and permission was granted. The next step in the process was to reach
out to the respondents for an interview, informing them that a study was being conducted
regarding math curriculum and instruction. Respondents were given information about interview
duration, time, and location. In addition, consent was requested for pictures of the classroom
environment, lesson plans, and student work samples. Table 2 outlines the participants who were
interviewed for this study.
Table 2
Participant Overview
Participant Grade Years of Experience &
Years of Experience at the
Specific School Site
Comfort Level in Math (1
to 4)
Respondent A 3rd

4 years teaching at
3 different school sites:
Located in Florida
Charter School A
Charter School B
Grade Level Experience:
1st and 2nd
4
Respondent B 3rd 6 years teaching at school
site
Grade Level Experience:
2nd and 3rd
3
Respondent C 2nd 4 years teaching at school
site
Grade Level Experience:
1st, 2nd, 5th
4
Respondent D 4th 6 years teaching at school
site
Grade Level Experience:
3rd, 4th, 5th

4

27
Respondent E 6th Long-Term Substitute at
school site
Grade Level Experience: K,
1st, 2nd, 3rd, 5th, 6th
3
Respondent F 4th 8 years teaching at school
site
Grade Level Experience:
3rd, 4th, 5th
4
Respondent G 5th 1 year teaching at school
site
Grade Level Experience:
2nd
4

Instrumentation
Since it is not possible to physically observe how people interpret the math curriculum
and their perspective, a semi-structured interview will be conducted with a variety of open-ended
questions and probes as needed (Patton, 2015). The semi-structured format allows the researcher
flexibility to respond to the situation (Merriam & Tisdell, 2015). In qualitative research, it is
important to make the interviewee feel comfortable so they can speak their truth, so a brief
introduction will be provided to provide context and reinforce anonymity. Document and artifact
analysis will also be used to study the physical material acquired in the study setting that is
relevant to the research questions (Merrriam & Tisdell, 2015).
For validity and reliability, triangulation will be used to confirm my emerging findings
through multiple sources of data, including interviews, artifacts, and documents. Triangulation
will be used to identify commonalities or discrepancies based on the different sources of data. If
multiple sources produce the same findings, they can be deemed to have a higher level of
validity. In addition, member checking will be used by taking tentative interpretations of findings
back to interviewees to ask if they are plausible (Merriam & Tisdell, 2015). The credibility of the
data speaks to whether the researcher presents data and findings that are plausible (Merriam &
Tisdell, 2013).

28
A semi-structured interview guide was designed to include a variety of questions times:
devil’s advocate, hypothetical, opinions and values, experiences and behavior, feelings, an ideal
that aligned to the themes, and research questions. Probes were included to seek more
information or clarification.
Table 3

Research Study Overview
RQs Themes Interview Document/Artifacts
What are the
characteristics of Eureka
Math?
Curriculum
● Essentialism
● Progressivism
● Constructivism
● Behaviorism

Q1–4 Eureka Math Teacher’s
Edition
What are the reasons
teachers modify Eureka
Math in their classrooms?
Teacher Ideology/ Teacher
Perception
Q5–8 Eureka Math Teacher’s
Edition
&
Teacher’s Lesson Plan
Teacher Content Knowledge
● KCS
● KCT
● SCK
● CCK
Q9–13 Teacher’s Lesson Plan
Teacher Self-Efficacy Q14–17 Eureka Math Teacher’s
Edition
&
Teacher’s Lesson Plan

Interview Guide
Introduction: Hello! My name is Christin Hwang. I am a Doctorate of Educational Leadership
student from the University of Southern California. I’m here to learn about math curriculum and
instruction. Thank you for taking the time to meet with me today. The purpose of this interview is
to understand the relationship between the curriculum, teacher, and instruction. There are no
right or wrong answers, or good or bad answers. I would like you to feel comfortable being
honest, saying what you think and how you feel. If it’s okay with you, I will be tape recording our
conversation since it is hard for me to write down everything while carrying on a conversation

29
with you. Everything you say will remain confidential, meaning that only I will be aware of your
answers. To begin, please tell me a little about yourself, including your education, your teaching
philosophy, and your experience.
Table 4 outlines the interview questions as they relate to the research questions.
Table 4

Interview Questions
Research Question Conceptual Framework Interview Questions Research
What are the characteristics
of Eureka Math?

Curriculum
● Essentialism
● Progressivism
● Constructivism
● Behaviorism
What do you believe are the
essential and non-essential
components of Eureka
Math?
Patton (2015)
Opinions and values
questions
Curriculum
● Essentialism
● Progressivism
● Constructivism
● Behaviorism

Some people would say
students’ interest is
irrelevant to math
instruction. What are your
thoughts?
Strauss et al. (1981)
Devil’s Advocate
Curriculum
● Essentialism
● Progressivism
● Constructivism
● Behaviorism

What do you believe is the
purpose of education?
Patton (2015)
Opinions and values
questions
Curriculum
● Essentialism
● Progressivism
● Constructivism
● Behaviorism

How do students learn best?
What are the dominant
reasons teachers modify
certain curricular
components of the Eureka
Math for their class?

Teacher Ideology Given the ELLs in your
classroom, which
components of Eureka Math
would you emphasize?
Patton (2015)
Experience and behavior
questions

30
Teacher Ideology Tell me about a time when
you had to deliver a lesson
for your student(s) with
special needs.
Merriam and Tisdell
(2015)

Teacher Ideology Please describe the students
in your classroom in the
context of math. What are
their strengths and
weaknesses?
Merriam and Tisdell
(2015)

Teacher Ideology In your opinion, does Eureka
math meet the needs of your
class and the individual
needs of your students?
Patton (2015)
Opinions and values
questions
Teacher Knowledge
● KCS
● KCT
● SCK
● CCK
What would your ideal math
curriculum/instruction look
like?

Probe: What would the
teacher be doing? What
would the student be doing?

Probe: What would be some
go-to instructional practices?
Strauss et al. (1981)
Ideal Position
Teacher Knowledge
● KCS
● KCT
● SCK
● CCK
Would you describe what
you think the ideal training
program would be like for
prospective math teachers?

Strauss et al. (1981)
Ideal Position
Teacher Knowledge
● KCS
● KCT
● SCK
● CCK

What do you think makes a
successful math teacher?

Probe: What skills do you
think are necessary?

Teacher Knowledge
● KCS
● KCT
● SCK
● CCK

Please take me through your
planning process using the
Eureka Math Curriculum.

Teacher Self-Efficacy Given your mathematical
abilities, which components
of Eureka Math do you feel
confident/less confident
about?
Patton (2015)
Feelings Questions

31
Teacher Self-Efficacy Some people say they are a
“math” person or an “ELA”
person. How would you
describe yourself?

Teacher Self-Efficacy Suppose you taught a group
of students who believed
they weren’t good at math
and didn’t like the subject.
What would you do?

Teacher Self-Efficacy Are you finding teaching
math as an adult a different
experience from how you
learned math?
Probe: How is it similar or
different?

Data Collection
Permission for access to teachers for the interview has been requested and was granted by
the principal at Corner Street Elementary School. An email was sent out to all TK–6th grade
math teachers at Corner Street Elementary with a brief overview of the study requesting
volunteers to participate in an interview after school hours in their classrooms. Seven teachers
volunteered for the interview. The semi-structured interviews were conducted one-on-one using
the interview guide in February 2019. With permission, the interview was recorded and notes
were taken by hand by the interviewer. The school site and all interviewees were given
pseudonyms to maintain anonymity.
Data Analysis
After completion of the interview, a review of the data was conducted to get to know the
data. Then the recording was used to provide verbatim transcriptions of the interview. Once the
data from the interview was transcribed, NVivo was used for data analysis. NVivo is a
qualitative data analysis computer software package produced by QSR International. It has been
designed for qualitative researchers working with very rich text-based and/or multimedia

32
information, where deep levels of analysis on small or large volumes of data are required. First
cycle coding was used to summarize segments of data using NVivo through nodes, then second
cycle pattern coding was used to group summaries into smaller numbers of categories, themes, or
constructs (Miles, 2014). Using the themes present, the researcher extracted assertions, big-
picture findings, and takeaways. The findings then informed the recommendations.
The data analysis was conducted by the primary researcher, who coded all the data into
themes or constructs. There can be some variation in themes or constructs that are identified
based on the data or how certain pieces of data are categorized to fit in a theme or construct
based on how the researcher perceives the data. Interviews were the main source of data used,
which did not allow for triangulation of data.
Summary
The curriculum alone does not guarantee high-quality teaching, but the teacher and the
curriculum shape each other in a dynamic interrelationship, and both are active participants
contributing to instructional outcomes (Charalambous & Hill, 2012). Remillard (2005) suggested
there is little research on how teachers use curricula or the teacher curriculum relationship. This
is a study of the intersection between the curriculum, teacher, and instruction in an elementary
math context. It seeks to understand the driving forces behind the instructional decisions of
teachers as they make meaning and interpret the curriculum to teach their students. Specifically,
it aims to uncover how and why teachers modify the curriculum by removing, modifying, and
supplementing certain components and their potential impact on student learning within the
context of a TK–6th grade school in South Los Angeles. This will provide insight into what
supports teachers require in pre-service and in-service contexts to provide quality math

33
instruction that increases students learning and performance because curriculum alone is not
enough.

34
CHAPTER IV: RESULTS
The curriculum alone does not guarantee high-quality teaching, but the teacher and the
curriculum shape each other in a dynamic interrelationship, and both are active participants
contributing to instructional outcomes (Charalambous & Hill, 2012). Remillard (2005) suggested
there is little research on how teachers use curricula or the teacher curriculum relationship. This
dissertation is a study of the intersection between the curriculum, teacher, and instruction in an
elementary math context. It seeks to understand the driving forces behind the instructional
decisions of teachers as they make meaning and interpret the curriculum to provide instruction to
their students. Specifically, it aims to uncover how and why teachers modify the curriculum by
removing, modifying, and supplementing certain components, and the potential impact of these
changes on student learning within the context of a TK–6
th
grade school in South Los Angeles.
Through the use of interviews, this study will examine teachers’ decision-making process as they
alter, adapt, or translate the curriculum offerings to make their instruction more “appropriate”
based on their knowledge, beliefs, perceptions, efficacy, and experiences.
This study will give us a better understanding of teachers’ decision-making processes in
modifying curriculum for instruction and its implications. This study would benefit teacher
educators and teachers themselves in understanding what misconceptions and needs teachers
have when modifying curriculum, which would ultimately benefit students. This will provide
insight into what supports teachers require in pre-service and in-service contexts to provide
quality math instruction that increases student learning and performance because curriculum
alone is not enough.
Research Questions
1. What components of Eureka Math do teachers emphasize or deemphasize?

35
How do teachers remove, adapt, or supplement certain components of Eureka
Math?
2. What are the dominant reasons teachers modify certain curricular components of
the Eureka Math for their class?
Conceptual Framework
To address the stagnant math performance across the nation, there have been efforts
around curriculum reform aligned to Common Core state standards. Unfortunately, curriculum
reform alone, without consideration of additional factors, has not made significant improvements
in student performance. The teacher’s role is to interpret the curriculum and provide instruction
to his/her group of students. The quality of the teacher’s interpretation of the curriculum has
consequences, too; Remillard (2005) stated that it would be inaccurate and irresponsible to
conclude that all interpretations of a written curriculum are equally valid. Even with the adoption
of a vetted, high-quality curriculum, how teachers interpret and deliver instruction is key to
student performance.
Franke (2015) observed that urban math teachers bring deficit views about students of
color that need to be challenged at the same time they transform content instruction because
math instruction for low socioeconomic status (SES) students of color emphasizes disconnected
concepts, math vocabulary out of context, following steps, and answers over explanations.
Teacher ideology and perceptions of the students they serve will influence the instructional
decisions they make as they deliver instruction.
Teacher knowledge and efficacy in math pedagogy and content are also contributing
factors to how instruction is delivered. Grossman (1990) named four different knowledge types
that are critical for teachers of math: common content knowledge, specialized content

36
knowledge, knowledge of content and students, and knowledge of content and teaching. Gaps in
any one of these four types of knowledge can impact the quality of instruction that teachers
provide. Strong knowledge of basic mathematical content in addition to the ability to represent
mathematical ideas in multiple ways or methods for appraising and evaluating mathematical
methods, representations, or solutions are important for teachers of math (Ball et al., 2008).
Content courses, professional development, or additional education that impacts a
teachers’ understanding of their craft supports their development of content knowledge and
pedagogy can be a valuable way to increase levels of self-efficacy (Swackhamer et al., 2009)
Highly efficacious teachers are more likely to use inquiry and student-centered teaching
strategies, while teachers with a low sense of efficacy are more likely to use teacher-directed
strategies (Swars, 2005). Tyler (1949) stated that learning takes place through the active behavior
of individual students—it is therough what they do that they learn, not what the teacher does.
Research suggests higher teacher self-efficacy in math positively impacts instruction through the
use of more student-centered instructional practices, which are more effective in math
instruction.
Knowledge of students and barriers between teacher and student are additional variables
that impact the delivery of the curriculum. Teachers teach a unique group of students who have
different interests, strengths, and areas of growth, with no two groups of students the same. As
United States classrooms have very diverse students from different cultures, Grossman (1990)
suggested that teachers need knowledge of content and students by understanding student
interests and common teacher misconceptions of students to effectively modify the curriculum to
meet the needs of their students. Progressivists such as Tyler emphasized the importance of
studying childen individually to find out what kinds of interest they have, what problems they

37
encounter, and what purposes they have in mind (Tyler, 1949). Limitations on time can also be a
barrier to the effective implementation of a curriculum.
There are a variety of variables that determine how teachers modify the curriculum,
including day-to-day barriers, teacher ideology, teacher efficacy, content knowledge, students’
knowledge, and pedagogical knowledge. Based on the teacher moves and modifications, the
teacher can enhance or decrease the rigor of the curriculum during instruction, which can have a
significant impact on student learning and performance. An understanding of how and why
teachers make instructional decisions will give insight into how teacher educators can approach
pre-service or in-service teacher preparation. Figure 1 provides an overview of the conceptual
framework.
Figure 1

Conceptual Framework

Teacher Curriculum Instruction
Barriers
Ideology/
Perceptions Efficacy
Content
Knowledge
Pedagogical
Knowledge
Knowledge
of Students
Student
Performance

38
Participants
A non-probability, criterion-based, purposeful sampling was used for the study in that all
teachers are currently math teachers in a TK–6th urban setting at Corner Street Elementary, part
of a large network of charter schools (LeCompte & Schensul, 2011). The convenience sample
was selected based on time, location, and availability. All teachers who participated in a semi-
structured interview at the same location over the course of three weeks. The seven teachers who
voluntarily participated in the study are purposive and convenience sampling that will allow the
researcher to discover, understand, and gain insight from information-rich sources from which
the maximum amount can be learned (Merriam & Tisdell, 2015). The teachers represent various
grade levels, years of experience, and comfort levels in math. Although the teachers are all
working at a charter school, the teachers have had experiences teaching in charter and public
schools, one with experience in a school outside of California. All teachers teach the same
curriculum, Eureka Math, which allows for comparison on how the same curriculum is
interpreted and carried out. Reference Table 2 for participant details.
Results of Research Question One
1. What components of Eureka Math do teachers emphasize or deemphasize? How do
teachers remove, adapt, or supplement certain components of Eureka Math?
Eureka Math is a widely used math curriculum aligned to Common Core state standards.
It has many characteristics that curriculum theorists deems important, including an effectively
organized group of learning experiences that include continuity (focus), sequence (coherence),
and integration (application) (Tyler, 1949). There have been many unsuccessful attempts to
address the underperformance of students in math through curriculum adoption alone. To

39
improve math instruction, one must first acknowledge the teacher’s role, then study the
instructional decisions made by the teacher to understand the reasoning behind those decisions.
Collopy (2003) found that teachers’ use of curriculum materials may be influenced by
their goals, interests, values, and expectations of the curriculum and may enact lessons in very
different ways than curriculum developers or educational reformers intended. The purpose of the
first research question is to uncover the components of Eureka Math teachers prioritize and
understand how and why they make instructional decisions when delivering the curriculum.
Theme One: Essential Components
There was consensus from all seven interview participants regarding what they deemed to
be essential components of Eureka Math: concept development, problem set (guided practice),
and exit tickets (independent practice). One respondent stated that the concept development is
where she gets the chance to give students feedback and address misconceptions, and that is
where the most amount of time should be spent.
There were differences in responses regarding some of the other components, including
fluency, workstations, application (daily problem solving), and student debrief. For example, one
person said that the component that she focused on less is the fluency portion because she didn’t
see its connection to a given day’s lesson or it was too difficult for her students. Another
respondent said she sometimes skips the application problem, which is the daily problem-solving
question, while another said they would alternate between fluency and application. A respondent
stated that it was important to give sufficient time for students to grapple with the application
problem; otherwise, it would just be another word problem they didn’t understand. A different
respondent said they knew the debrief is important, but is often difficult to include due to time
constraints. Since the debrief portion of the curriculum requires the use of academic discourse,

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the respondent stated she adapts the curriculum to incorporate academic discourse into the
concept development and the problem set. She stated that the fluency and application
components were more flexible, whereas the conceptual development, problem set, and exit
ticket were integral.
The fact that majority of the teachers believed the essential components of Eureka Math
to be the concept development, problem set, and exit ticket aligned to the model of direct and
didactic instruction: I do, we do, and you do. There are elements of different curriculum
philosophies that are reflected in the concept development, problem set, and exit ticket. The
essentialist or traditionalist theory states that there is specific knowledge that has been deemed to
be important to learn based on the past. The Eureka Math curriculum has determined specific
content and concepts that need to be learned based on their interpretation of the Common Core
standards. Further, the essentialist or traditionalist theory states that learning takes place through
the experience the learner has which the teacher provides, and the attitude of pupils must be
docile, receptive, and obedient (Bagley, 1941). The curriculum does also have elements of
constructivist cognitive theory in that the conceptual development portion has students develop
their understanding through concrete, pictorial, and abstract means, which align to enactive
representation (action-based), iconic representation (image-based), and symbolic representation
(language-based) (Bruner, 1961). Tyler (1949) stated that there are three criteria for learning
experiences: continuity, sequence, and integration, which is described in the Eureka Math
curriculum as focus, coherence, and application. Elements that are not found in the concept
development, problem set, or exit ticket are elements of progressivism, where the curriculum
emphasizes the importance of studying children to identify their interests, problems, and
purposes, with the belief that childrens’ interests must be identified so they can serve as the focus

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of educational attention. There is also a lack of social reconstructionism, with no emphasis on the
concerns, interests, and injustices students identify as the starting point for meaningful and
worthwhile learning.
There was also consensus on the components of Eureka Math that were followed more
flexibly: fluency, application, and student debriefs. All seven respondents identified at least one
of these components as being flexible, which they saw as warranting decrease in frequency or
duration of implementation. For these components, all seven teachers were inconsistent with
delivering all of these components daily. Four out of the seven respondents stated they would do
either fluency or application on a daily basis, instead of both. These components were also
sometimes adapted to be integrated into the concept development, for example, incorporating
discourse in student debriefs in the concept development. The flexible components as identified
by the respondents are considered the non-essential components of the Eureka Math curriculum.
Three respondents stated there were essential components of instruction that were not
explicitly included in Eureka Math, such as explicit vocabulary instruction, group work, and
sentence frames.
Theme Two: Teachers ’ Role
Across all of the seven respondents’ interviews, a common theme that arose was that
teachers had a role in using the curriculum to provide instruction. They believed it was their
responsibility to provide beyond what was provided strictly from the curriculum, and these
additional methods encompassed creating structures for math instruction, intellectual prep work
before the lesson, and instructional strategies during and after the lesson, including
differentiation and scaffolding, teacher engagement, and role in student engagement.
Respondents identified many moves as being part of teachers’ responsibility in implementing the

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curriculum: adding excitement to engage, celebrating mistakes, incorporating games, focusing on
vocabulary, providing visuals, providing manipulatives, creating partnerships, providing
feedback to increase confidence, facilitating small group instruction, providing choice, creating
anchor charts, and color coding. One teacher stated, “If you are reading the script and expecting
kids to respond the way it’s scripted, then you will lose most kids.”
Structures
In an interview, one respondent stated that a foundational piece necessary for a successful math
environment that teachers believe they need to provide is to “set up routines and procedures in
the classroom.” Good classroom culture is needed before teaching. Additional structures such as
“accountable talk, or partner work” also needed to be explicitly taught in how students engage
with each other.
Planning
Planning was a common theme that arose across three interviews. This included the need
for unit planning as well as daily lesson planning and what teachers believed they needed to do
to deliver the curriculum. One respondent described the unit planning session with collegues and
dean of instruction, which started with identifying the standards and reading through the
overview to understand how the curriculum is written and what are the author’s intentions behind
the unit. During this planning session, they determine anything that needs to be modified or
added as well as the additional resources that will be used. There is also consideration of the
strategy and method provided in the curriculum and if what was suggested is the best approach.
Teachers look at the exit tickets for the module, identify misconceptions, identify unfinished
learning, and begin looking at the lessons to find ways to customize it if the way the lesson is
written by Eureka Math would be “tricky” for students to understand.

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The daily lesson planning focuses on the exit ticket as a means to backwards plan,
identifying the criteria for success, then reviewing the concept development. The problem set
question is rarely assigned in its entirety and is specifically chosen to align with the exit ticket.
Two respondents shared that they collaborate and go through this process with their partner
teacher weekly, going back to the standards for the lesson, understanding the vocabulary,
identifying the criteria for success for that standard, and determining prerequisite knowledge.
Theoretically, the unit planning and daily planning process allow teachers to internalize
the content, make connections between lessons by understanding the progression, and make
informed decisions to supplement and modify the curriculum, with the knowledge of where the
lesson will lead them, and thus where to go next.
Knowledge
Five respondents discussed that with the changes in Common Core math, the need for
“knowing” the math by “learning all these new ways of solving a problem so that students can
learn it as well.” To build knowledge of the math content, one respondent stated she spent hours
doing the problem set on her own because she wanted to make sure she understood it so that her
students understood it. Respondents also discussed how the teacher’s role is to understand the
content and pedagogy to fit their class. Both of these responses refer to the content knowledge or
the pedagogical knowledge needed in math instruction. As one interviewee reported, “I think that
a good successful math teacher needs to know what they are teaching. To know the ins and outs
of it. And also, to explain it in different ways.” One respondent referred to the curriculum more
as a guide than a script: “It is always there for you to use, but my role is to kind of see what
components fit my class and pick and choose what is right for them, or what is their need. I think
that the role as the teacher is just for you to take all this in and try to figure out what best suits

44
your class.” They discussed having to rework, condense, and cut out specific things due to the
language or the mode of presentation to get students to understand the standard and be able to
grasp the main concept or core.
Teacher engagement. One teacher stated the teacher’s demeanor plays a huge part in
how motivated they are to learn math. Especially with a program like Eureka, the way that the
teacher approaches it is vital because, as mentioned above, “if you are reading the script and
expecting the kids to respond the way that it’s scripted, then you're going to lose most of the
kids.” Another teacher spoke to the importance of the teacher’s demeanor and attitude towards
math: “So, you just need to know math and actually if you don’t love it, pretend to love it. I love
math but some people don’t.” A teacher stated, “If they’re interested and they see that you are
excited, even the kids who don’t like math or don’t think of themselves as math people will be
like, ‘Oh, wait, this strategy, I can do this.’”
Teacher s’ Role in Student Engagement
Four respondents discussed the importance of building relationships with students to get
to know them, who they are, their interests, their special needs or accommodations. A respondent
stated, “I mean I feel like student interests should always be a part of your lessons, you always
need to tie it into their life, their experiences and just what they connect with. I feel like for math,
especially with the curriculum, you have to be very strategic and find those places where you can
include the student interests. Because the curriculum doesn’t allow for that.” Other responses
also shared the need to make the curriculum relevant and incorporate things students can connect
to increase engagement which ultimately improves their learning. Another respondent stated to
create buy-in she tried to bring in their prior knowledge of what they know about the subject.

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One person discussed the value of ensuring students are comfortable in the classroom and
developing a self-awareness as teachers that they believe in their abilities because this would
allow them to take on greater challenges. In addition, some respondents discussed the importance
of taking ownership of students’ learning and reflecting on their practice as a teacher and
providing differentiation because all students are different. Additional strategies mentioned to
increase student engagement was to “make it a game,” incorporate music and movement, sing
songs, and make the content more “fun.”
Three respondents discussed the role of the teacher is to make a connection with the
content to the real world to increase student engagement. “I think I would try to a) have them see
the purpose of math or like why, you know, it’s important, ’cause I mean I’ve had a lot of
students that it’s like, ‘Why? I don’t need this. I don’t need it.’ So, I think first doing that and
then trying to break things down where they could be immediately successful and go slowly with
them but not too slow and provide a lot of positive reinforcement. You know, ‘you can do it’ and
stuff like that.”
T e ac h e r s ’ Role During and After the Lesson
All seven respondents stated that teachers had to make professional decisions and
modifications to the curriculum to allow students to conceptualize the concepts as opposed to
remembering several procedural steps in an outmoded approach to math instruction. There was a
belief that one skill necessary to be a successful teacher is being able to make adjustments to
your lesson. Teachers make adjustments before implementing the lesson during the planning
process and during instructiona as they gather formative data.
Three respondents also discussed the importance of feedback to improve understanding,
self-monitoring, and confidence. As one explained,

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When we do problem set, we always come back and check it before we debrief so that’s
not part of Eureka, but I feel like, for my kids, they need to know like, “Am I on the right
track? Was I getting this?” to build that confidence and then, I find that that’s helpful to
bring them right back before we debrief. I feel like kids learn best if they have feedback
on what they’re doing. I always think about how I learn best. I need to just jump in there,
try, of course, you always wanna give them some direction but you wanna let them
explore and make their mistakes and grapple and feel the frustration but guide them
through it.

Three teachers also specifically discussed their role to use data to inform instruction. “For
me what I decide is I do take in their exit tickets, even though I try to correct it the same day. I
think it’s really important for me to realize what they still need help and the misconceptions that
they may have, and be able to do that.”
Differentiation
A respondent gave an example of how she provides differentiation and choice for
students: “After I teach my concept development, I have the students decide whether they are
going to work on their own or whether they need extra help. So, I give them choice. So, they stay
in the carpet or I do a small group here at my desk. So, for those students I use manipulatives a
little bit more. I sometimes draw, kind of grasp they might need, kind of like an organizer to
organize their math information. While everyone is working in that time, I usually will have that
student sit with me and I will either guide them with manipulatives or just break it down into
pieces and try and use, you know, different visuals.”
Another teacher shares how she differentiates for EL by color coding anchor charts:

For the ELLs, I try to do a lot of sentence stems and visuals and manipulatives.
Manipulatives for them as well. I feel like those are essential for all kids though, not just
for ELs. So, for the ELs, we had to include sentence frames, especially when we were
answering those application problems. I try to incorporate what the image is in their
primary language, or what the word is in their primary language and just the same
picture. That is pretty helpful when it comes to Eureka math because some of the word
problems are difficult.

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Respondents also pointed out the use of supplemental materials such as whiteboard,
manipulatives, anchor charts, explicit vocabulary instruction, note-taking systems, and partner
talk to provide more support and differentiation within the whole group needs, subgroup needs,
and individual needs. Within the whole group context, a majority of respondents discussed the
need to be able to use formative data to determine the next steps rather than checking off lessons
to be taught. In addition to the regular curriculum provided, respondents also discussed the need
for small group instruction.
Theme Three: Teacher Development
Interview data suggests quality teacher development in math instruction needs to address
the why, what, and how. In addition to content and pedagogical knowledge, well-prepared
teachers need to make connections between the CCSS and the curriculum, understand the
curriculum, then deliver the curriculum.
Why
Three respondents stated that it’s important for teachers to understand the bigger picture
and the curriculum’s connection to the Common Core State Standards: “They need to understand
why they are teaching something, why they are teaching it in a specific way, and how it all
connects to the bigger picture.”
What
Respondents called for specific curriculum training for teachers that encompasses more
than observing other teachers. They found it necessary to dig deeper to understand how the
curriculum was structured and how it was written and organized. Through the planning process,
they wanted to see the connection module by module, the connection between lessons within a
module, and the connection to standards.
How

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Four respondents also stated there was a need to understand how the lesson on paper
would be executed in the classroom and taught to a group of students. One suggestion that came
up was the concept of teachbacks, which would allow teachers to teach the lesson to their
colleagues or instructional coaches, then get feedback before teaching it to their students. One
respondent stated that seeing a demonstration lesson as well as an opportunity to practice the
lesson before teaching it in front of students were beneficial. They also discussed how getting
feedback from someone else is helpful to prepare for the delivery of a lesson.
One respondent stated,
I think that a lot of teaching programs focus a lot on the whys but not like, “This is how
you deliver it, this is how it’s going to be.” We’d be scripting, how it would sound like
and feel like, watch a lesson demonstration and having someone watch you, to provide
feedback. Also, having a chance to do it sooner allows you to see what comes up and
how you might attend to it at the moment when explaining certain concepts.

There were some conflicting responses, as when one respondent stated, “I think the ideal
training would be teaching the components of the curriculum, how to look at it, what things
could be altered and discussion with grade-level teams on what to do. I’m not huge, let’s teach
each other the lesson. I don’t find that beneficial. I do find it beneficial seeing somebody do it, so
like an example.” Overall, there was consensus on seeing how the lesson “should” be taught and
mixed responses regarding whether teachers wanted to engage in teachbacks with their
colleagues.
Follow-up Support
Five respondents shared the need for continued support using a curriculum. For example,
they requested guidance on which portions of the curriculum are mandatory. One respondent
stated, “It would be great to have someone observe you and give you feedback on what you need
to improve, some glows and grows because I think that a lot of our teachers feel overwhelmed

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with Eureka.” This demonstrates that continue professional development and follow-up support
is necessary for the implementation of a curriculum.
Discussion of Research Question One
1. What components of Eureka Math do teachers emphasize or deemphasize? How do
teachers remove, adapt, or supplement certain components of Eureka Math?
Based on the qualitative interviews, teachers identified the essential components of Eureka Math
to be concept development (I do), problem set (we do), and exit ticket (you do). Based on what
they identified during the interview as being essential and what was identified on their lessons
plans, teachers preferred the didactic instruction of identified essential content where they were
the knowledge providers and students were expected to learn what they are taught. Teachers did
supplement in these areas based on their perceived needs of the students. Supplementation
included explicit vocabulary instruction, group structures, and differentiation. Based on
constraints, teachers deemphasized the application, fluency, and student debriefs components of
Eureka math. Depending on the teacher, some of the deemphasized components were removed
and or adapted.
Teachers did find it important to supplement components of teaching that they deemed to
be important in the content and delivery of instruction and viewed it to be part of their role. This
included planning for instruction before, during, and after instruction beyond what is present in
the script, and preparedness in terms of content knowledge, engagement, and differentiation.
Teachers also discussed the importance of teacher development through professional
development, coaching, and collaboration, effectively removing, adapting, and supplementing
elements of the curriculum. It was specifically mentioned that this type of support should be

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continuous and there should be follow-up rather than a “one-and-done” approach. The focus of
this session would be to understand the why, what, and how of the curriculum.
Although there was a variety of responses based on the interview guide as teachers
interpreted the curriculum differently and there was a range of responses regarding essential and
non-essential components of Eureka Math, these were the main themes that arose.
Results of Research Question Two
2. What are the dominant reasons teachers modify certain curricular components of
the Eureka Math for their class?
The purpose of this question is to understand how and why the teachers modify certain
curricular components of Eureka Math. Reichenberg (2016) defines text competence as the
teachers’ ability to use textbooks strategically. Even when teachers receive a high-quality
curriculum aligned to CCSS, they must understand the curriculum and interpret its meaning to
provide instruction. There is a continuum of approaches across teachers using the same
curriculum. Once there is an understanding of how and why teachers modify curriculum, and the
impact these changes have on student experience and learning, this information can be used to
prepare teachers pre-service or in-service for effective math instruction.
Theme One: Barriers

Time constraints were a theme across many interviews as a reason why certain
components of Eureka Math were condensed or cut out as teachers prioritized what they deemed
to be most or more important. Some teachers stated they didn’t understand the math, which
speaks to the teacher knowledge gap. Akkus (2016) stated that less than half of elementary
teachers felt well-prepared for teaching math, compared to 60% of middle and 70% high school
teachers. This speaks to the fact that most elementary teachers do not major in math, and with the

51
shift to CCSS, content knowledge has become even more critical in elementary math instruction.
Teachers are required to play a substantially different role in math classrooms than ever before in
regards to mathematical thinking, conceptual understanding, and problem-solving. Howe (1999)
suggested Chinese teachers have a much better grasp of the mathematics they teach than
American teachers. For example, 20% of American teachers had a conceptual grasp of
regrouping vs 86% of Chinese teachers (Howe, 1999). One teacher stated:
I was not confident at all about the concept development because you as the teacher are
guiding their development of a concept. If you don’t understand where you’re going to
end up and what the purpose of that is, there’s no way you can guide students to get there
without just telling them. Being able to understand where you’re going not only within
that lesson but within that topic, within that module is so important for you to be able to
gauge, “Where are my students now and how am I going to guide them to this concept?”

This response showed teachers’ lack of pedagogical content knowledge and content knowledge
to inform their teaching practice. The lack of knowledge also negatively impacted teacher
efficacy. Teachers with a low sense of efficacy are more likely to use teacher-directed strategies
versus inquiry and student-centered teaching strategies (Swars, 2005). Research suggests highly
efficacious teachers are more effective mathematics teachers than teachers with a lower sense of
efficacy (Swars, 2005). Teacher math content knowledge, pedagogical knowledge, and self-
efficacy all play a role in the math instruction American teachers are providing in classrooms.
Respondents stated another barrier was students’ lack of interest in Eureka Math, leading
to low motivation. According to Elmore’s (1990) theory of the instructional core, this
demonstrates a lack of consideration of the student. Elmore (1990) describes the instructional
core as the intersection of student (engagement), content (rigor and relevance), and teacher
(knowledge and skill), with the task (what students are doing) at the center. Although the Eureka
Math curriculum may have the relevant and rigorous content embedded, it does not consider the
students, engagement with whom is a critical part of the instructional core.

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Teachers discussed the need to make changes to the curriculum due to the amount of
content and lack of time but discussed a lack of confidence in tailoring the curriculum,
specifically the concept development. For example, one respondent stated, “I wasn’t tailoring the
concept development to my students, and then when I was tailoring it, I didn’t know how to
tailor it so that I wasn’t getting rid of the rigor and I wasn’t getting rid of the students doing the
thinking part.” The respondent stated that due to being unable to understand some of the content
within the curriculum, she would cut it out or tell students something different: “A lot of times, I
would go back to ‘I don’t know why they’re asking this so I’m going to just tell them this.’” This
speaks to the teacher’s knowledge and skill component of the instructional core. The curriculum
itself is unable to control for teacher knowledge, and a change in any component of the
instructional core impacts student learning.
Another driver for making changes to the curriculum was based on the belief that their
struggling students needed more scaffolds. When examining the curriculum, the teachers also
perceived the exit ticket (independent practice) was more difficult than the direction and guided
instruction. This speaks to the teacher’s educational philosophy of having students copy or
practice what they are taught rather than generalize, apply, and expand on what they are taught.
One respondent stated, “Ideally with assessments at the end we would have a check-in and exit
ticket, and then the assessment would be almost exactly what we did, not something different.”
This viewpoint accords with the didactic approach of teaching. Teachers expected students to use
the same process and methods they had used to solve the problems when they were working
independently instead of generalizing, applying, and expanding what they learned.
The curriculum also included different strategies for solving questions and teachers chose
to cut out strategies that were less familiar to them. One respondent stated:

53
I know something that I was not super comfortable with was the Arrow Way. It’s because
when the students sit down and are writing so much, and I remember that we were doing
two-digit addition and subtraction and Arrow Way was one of the strategies, and I was
like, “Oh, I kind of don’t like these strategies.” So, I didn’t focus on it a lot. And then I
was planning for my next lesson is when I realized that Arrow Way was one of the ones
that they mainly focused on with three-digit numbers, and I was like, “Oh, man, I really
should have spent more time teaching this strategy.”

The math strategies used in solving math in Common Core math were different from how
teachers learned math. There was a level of discomfort using the newer strategies, especially
when teachers did not have the conceptual understanding and “why” behind that strategy:
It’s different because it’s not how I learned it in terms of Common Core and
decomposing numbers and playing with numbers. In school for me, my memories of
math growing up was very like, this this this, this this this, this this this. It wasn’t really
like seeing numbers as parts of other numbers. Do you know what I mean? So, it is
different in that way. It’s similar in that, you know, it’s universal and so it doesn’t really
change, but growing up it wasn’t like now where you are given all these different
strategies to learn it and then whatever works for you is best. Whereas, it was just like
this is the way to do it. Growing up it was like, this is how you do it, that’s it, there’s no
other way. Yeah, and I find that some of the things that we teach now are things that later
on in life I just figured out how to do on my own with the play of numbers, like go
finding tens or finding twenties or growing up it wasn’t like that. And I think that’s why
it was really hard for me to understand Eureka when I started, ’cause it was completely
different from the way I learned math.

Teachers chose to make changes to the curriculum due to their comfort level and previous
experience in the content. This has serious implications because they may be eliminating critical
components of the content such as important strategies that they will need to use in future
lessons. Also, because they didn’t have the insight of the curriculum writers and the bigger scope
in sequence, it was difficult for them to anticipate the significance of certain content and
strategies and how they would impact future lessons.
Another variable to consider is the connection between planning and instruction. The
planning is the intention, whereas, the instruction and students’ learning are what actually
occurred. There is frequently a gap between the intention versus what happens, or between

54
planning versus instruction. This gap also occurs between the curriculum writer’s intent versus
the reality of day-to-day instruction and how it is carried out. “What I’m trying to say is like
you’re teaching all these strategies and sometimes it gets all mixed up. And you kind of lose the
students, especially with how fast-paced it is. When we were teaching one strategy, then the next
day it’s a different strategy. Now both strategies, and three strategies and it’s just so quick.
That’s sometimes is hard for students to catch on. So I understand why, what the intent was, but I
think it’s just really hard.” Teachers are grappling with understanding the intent, delivering the
instruction, and responding to students’ needs.
Theme Two: Content Knowledge
Students
Respondents made changes to the curriculum due to the perceived knowledge gaps they
identified in their students. Respondents identified fluency ( addition, subtraction, multiplication,
and division facts) as an area of struggle for most students. They also identified word problems
as being difficult for students due to their struggles in reading. This prompted some teachers to
remove the application problem since they anticipated it would be too difficult for students. One
respondent stated, “I know that word problems can be difficult for them so sometimes I feel least
confident in making sure that they get that application problem.” The application and word
problem are more rigorous than the problem sets, and when they were left out of the lesson,
students did not have an opportunity to engage in higher-order thinking tasks. Franke (2005)
stated that urban teachers bring deficit view about students of color, which is reflected in math
instruction that focuses on following steps and over explanation.
Teachers

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Teacher knowledge specific to Common Core math is not addressed within the
curriculum and has an impact on the delivery of instruction. All seven respondents stated they
did not learn math in a way that focused on conceptual understanding, and their experience with
math instruction was mainly procedural with an emphasis on using the standard algorithm. A
respondent stated, “I remember that as a student, especially at the elementary level, it was a lot of
memorizing stuff and not understanding the why or the how.”
As part of the intellectual preparation, teachers felt they needed to invest the time to
understand Common Core math. Once respondent stated, “I had to read the curriulum over and
over to try to pull out the points that I needed to figure out so I could teach the lesson to my kids,
make sure I was asking the right questions and I understood the objective. I would spend hours
doing the problem set on my own because I wanted to make sure I understood it so that my
students understood it. I had to learn math all over again because the math was not familiar.”
Theme Three: Ideology
Respondents had a variety of beliefs when it came to their ideology and philosophy
around education. A respondent stated the purpose of education is to make productive citizens in
the world, to prepare kids for what “real life” is going to be. Some spoke specifically about the
context of the students and community being served: “Especially in our community, I feel like
it’s even more of a responsibility because our kids are going to have so many more challenges
than students in more affluent communities are never even going to experience. Yes, to prepare
them in terms of education, but also in terms of helping mold good citizens, kind people.” One
respondent stated the purpose of education is to teach life skills too: to learn, to problem solve, to
be like an investigator, to be reflective. The purpose of education is to prepare kids for college,
life, and society. “They need to know how to read, how to do the math, and how to write.” Other

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respondents had similar beliefs and stated the purpose of education is to teach the whole child, to
teach children to be problem solvers, to think for themselves, to appreciate mistakes. Others said
that the purpose of education is to gain knowledge, to apply either at a job or in a career.
There was an emphasis on classroom culture as part of learning. “If they’re comfortable
in the classroom, I feel like that’s how they learn best, then if the way that they are being taught
also speaks to their needs is how they learn best.” One respondent stated that it was critical to
have clear and high expectations, building a classroom culture where everybody feels valued and
loved. Teachers believed in high expectations for the students and provided a scaffold for them
to give them the supports they needed. One respondent stated that her philosophy changed
throughout her career. She stated, “I feel like as a new teacher, I feel like I had control over the
classroom to feel successful. Now I feel like the kids should be thinking, should be leading the
class, and a student-centered approach is important.”
Three respondents discussed looking at students as individuals and meeting students
where they are: “I try not to do blanket teaching if that makes any sense? Like teaching
everybody the same, because a) they are at different levels and b) everyone learns differently. I
really wanna meet them where they are and build on that.” Another respondent discussed the
importance of being culturally responsive: “I really wanna validate them and affirm them and
make them believe that they can be successful and that education is important because it will
provide a choice for them. A choice to be whoever or whatever they want when they’re older.”
Three respondents believed students learned best from each other, working
collaboratively. One specific example was that groups are better for them because they are able
to scaffold each other. Two respondents discussed the importance of allowing students to grapple
with the information since that is when “real” learning occurs. This involves allowing students to

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make mistakes and help them understand that they have to persevere. Teaching math also
incorporates life lessons along the way.
When trying to uncover why teachers made changes to the curriculum, it’s important to
understand their ideology and beliefs when it comes to education. Their ideology reveals what
they value, why and what they do to make revisions to the curriculum. For example, the
curriculum itself doesn’t integrate any collaborative groups, but some teachers chose to build
those pieces in because they align to their teaching philosophy.
Theme Four: Ideal Math Instruction
When teachers were asked what their ideal math instruction consisted of, there were a
variety of responses focused on engagement, problem-solving, collaboration, small group
instruction, and use of manipulatives.
Engagement
Making instruction engaging and fun was a priority for five respondents. Teachers
wanted students to be actively engaged, taking on a greater role and doing more talking. Two
respondents referred to a balance of teacher talk versus student talk, preferring students to
engage more. As one explained, “the teacher probably would be doing less talking than what I do
now because I feel the students are super quiet and they don’t wanna say an answer because they
think they’re gonna be wrong. I would love if we had more student engagement.”
In addition to making instruction fun, engagement also requires the teacher to notices that
they’re losing their students and so tries a different explanation. This respondent explains why
it’s important for teachers to have an understanding of the content in order to keep the students
engaged: “You don’t have to be a mathematician, but you have to understand the concept you’re
teaching at that moment so that you can stop the lesson and say, ‘Okay they’re not getting it. Let

58
me give them a different analogy so they can understand what I’m talking about.’ If you can do
that at a moment’s notice, just adjust, then do that, and be successful.”
Role of Teacher and Student
Three respondents discussed how the role of the teacher is a facilitator. A respondent
stated, “I would be the facilitator and the students would be working on tasks in partners or
groups and they would be working towards the same goal, but they would be showing each other
how they got to the same goal but in a different way. And there would be a lot of student talk and
building on each other’s conversations.” Another respondent stated, “I would give more time for
little kids to try it on their own and make those mistakes, and just persevere through the problems
because I feel like a lot of them that’s what they need.”
Problem-Solving
Four respondents brought up having both authentic application problem-solving tasks and
supporting students in becoming problem solvers. For example, one respondent said she wanted
to support students in being problem solvers instead of just getting the right answer by providing
real-world problems, more opportunities for problem-solving, and more time to problem solve:
I would pose a problem and then have them grapple and try to solve it and come up with
different ideas. And then I would . . . let’s say, that this would be the application problem,
or like a big problem that’s kind of that they are intrigued and they are engaged in. And
then after that, I would teach a concept and have them do it again. And then find the
answer and it would be a teaching moment and then I would teach. Well, this is what I
just taught you. And then we would go into the problem set and have them practice. And
then at the end debrief, go through the questions. Project-based learning, where we would
start on a concept and build on it, but in the end we would have produced something. So
it was not only hands-on, being related to something that was a real-world experience.
And they produce something. We are currently teaching area and perimeter . . . I don’t
know, like let’s build a little house or something. They would measure things and how
much would it cost. Something like that. And give collaboration time for students to
collaborate.

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One respondent specifically shared the positive impact she has observed. “I’ve seen a huge
increase in the ability to reason why they’re doing something instead of just how to do it and
then embracing multiple strategies.” Some students were able to build their conceptual
understanding through problem solving allowing them to make connections between what, how,
and why they are doing something. This is part of some respondent’s ideal math curriculum and
instruction because it allows students to develop an understanding of concepts and draw on what
they know to flexibly solve problems in the most efficient and effective way.
Small Group Instruction
Five respondents discussed small group instruction as part of being their ideal instruction.
A respondent stated,
My ideal instruction would be just all of it in small groups. A guided math structure, so
one group is here, I teach you the concept development here, I do the problem set and
then go and take your exit ticket and then I get the next group and it rotates and then all
the other kids are, so it is like stations while I’m teaching the actual math lesson as
opposed to the whole group. I would be doing the same thing that I would do the whole
group, except things with manipulatives would be a lot easier to use, it would give me a
little more time and it would be easier for me to see immediately who’s not
understanding so that I can switch things around right then and there as opposed to I’ve
taught everybody, go do your problem set and let me check your problem set and then
take your exit ticket, and give me a lot quicker just formative data.

Manipulatives

Four respondents discussed the importance and benefit of using manipulatives. One
respondent said, “Giving them those manipulatives and even real-life scenarios that they can
practice in the class will allow them to experience math and understand the purpose of math.”
The challenge identified in using manipulatives was limited time.
This theme speaks to what teachers value, which may provide information as to what
teachers will do in their classroom. The responses to this question about their ideal math
instruction shed light on their ideology and its impact on their instruction. When using the same

60
curriculum, do teachers change their instruction to match their ideology and ideal math
instruction? Engagement, teacher role, problem solving, small group instruction, and
manipulatives were identified as being important components of math instruction. Some of these
elements were integrated into their teaching to different degrees and represent how teachers
adapted and supplemented the Eureka Math curriculum.
Discussion of Research Question Two
2. What are the dominant reasons teachers modify certain curricular components of
the Eureka Math for their class?
Based on the responses from the interviews, teachers modify curricular components of
the Eureka Math for their class. The curriculum is interpreted and implemented based on barriers
that are present, current content knowledge, and teacher ideology. Based on the various barriers
present, certain parts of the curriculum were prioritized. With time constraints being a barrier,
teachers implemented components of the curriculum they valued and deemed more “important.”
Another barrier identified by respondents was the varying levels of content knowledge and
pedagogical knowledge teachers possessed. Teachers also stated there was little consideration of
student interest in the curriculum from Eureka math and its designers, which led to a lack of
student engagement and low motivation. They also identified a gap between their students’ needs
and the curriculum as a barrier that needed to be addressed. For example, one teacher stated
Eureka does not include a lot of vocabulary instruction or ways to incorporate writing into math,
and that is something that English learners struggle with a lot.
Another reason teachers modified the curriculum was due to the perceived gaps of
foundational knowledge their students had. They identified unfinished learning in their students,
which they felt needed to be addressed to be successful in lessons of the current curriculum. In

61
addition, they identified certain components, such as the application problem, as being difficult
for their students because they struggle with word problems.
Teachers also made changes to their curriculum based on their ideology and what they
deemed the ideal math curriculum included. Their view of the purpose of education and effective
pedagogical practices shaped the way they supplemented the curriculum.
Summary
Instructional Core
Based on interview responses, the most emphasized component of the Eureka Math
curriculum was the concept development, problem set, and exit ticket. This supports the didactic
approach and the lesson structure: I do, we do, and you do. When implementing the curriculum,
teachers perceived their role to have multiple responsibilities before, during, and after teaching
the lesson. Teachers identified the need to set up structures for teaching math and their role in
planning, teaching, and reflecting on the lesson.
The reason why teachers made changes to the curriculum was due to a variety of reasons
including time constraints, teacher knowledge, teacher perceptions, and teacher ideology.

62
CHAPTER V: DISCUSSION
Findings
First, I wanted to understand the different components of Eureka Math and how they
aligned to the approaches identified by curricular theorists. Then, I wanted to identify which
components teachers emphasized or deemphasized by identifying the teacher moves of
removing, adapting, or supplementing elements of the curriculum. This would shed light on what
components of the curriculum are teachers modifying and the implications of their actions. Do
the modifications alter the curriculum in ways that negatively impact student learning, according
to curricular theorists?
The interviews revealed that certain curricular components of Eureka Math were
emphasized and prioritized, while others were de-emphasized, reduced, or eliminated. In
addition, some teachers supplemented the curriculum. Teachers identified the core components
of the curriculum to be concept development, problem set, and exit ticket, showing they favored
a more didactic approach to instruction as it followed an “I do,” “we do,” and “you do” model.
Respondents also identified the application, fluency and student debrief components as
being “less important” and more flexible. The application component is the problem-solving
portion and the student debrief provided academic discourse opportunities.
Findings of Research Question 1: What components of Eureka Math do teachers
emphasize or deemphasize? How do they remove, adapt, or supplement elements of the
curriculum?
Finding 1: Emphasis on Concept Development (I Do), Problem Set (We Do), and Exit Ticket
(You Do)

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In Eureka Math, the concept development is the lesson portion where new learning takes
place. All teachers prioritized the concept development and followed up with students doing the
problem set and exit ticket. They did make modifications in the number of questions students
completed in the problem set. Some stated that they would have students do the problem set or
exit ticket, but all respondents stated it was important for students to solve problems similar to
the concept development with support or on their own. Based on the components of Eureka Math
that were emphasized, the teachers favored a more didactic approach and provided mostly direct
instruction.
Finding 2: Deemphasis on Application, Fluency, and Student Debrief
Another finding was that other components of the curriculum were deemphasized, such
as the application problem, fluency, and student debrief. With time constraints, teachers
eliminated, shortened, or implemented these components inconsistently.
The application problem within the curriculum can be considered problem-solving, which
in the context of math involves word problems. These allow students to make connections
between the math they are learning and a real-world context. The fluency components support
identifying patterns and memorization, for efficiency. The student debrief is an opportunity for
academic conversations, with students identifying and discussing strategies, misconceptions,
patterns, and making meaningful connections. With the student debrief taking place towards the
end of the lesson and without a concrete and tangible output such as an exit ticket, it was
shortened or removed, especially during time constraints.
Research: problem-solving, application, real-world problems, academic conversations
Finding 3: Supplementing the Curriculum: Engagement Strategies, Language/ Vocabulary,
Real World, Examples, Collaborative Work Structures

64
Based on the interview responses, it was apparent that teachers felt that the curriculum
was missing many components that needed to be supplemented. Teachers stated that the
curriculum lacked engagement strategies in how the curriculum was delivered. The curriculum
included the content to be taught but lacked the pedagogical components. The curriculum
included the “what” but not necessarily the “how,” including structures for collaborative work.
They also stated that explicit instruction in math vocabulary was absent in the curriculum.
Although the curriculum referenced the key vocabulary words in context, it failed to include
explicit instruction. Given that a large portion of the students they work with are English
learners, this was something they felt was critical to be included in the curriculum.
Findings of Research Question 2: What are the dominant reasons teachers modify certain
curricular components of the Eureka Math for their class?
Finding 1: Time Constraints
It was clear that the users of the curriculum interpreted and implemented the curriculum
in different ways, making modifications. One of the top reasons for making modifications was
time constraints, which necessitated cutting out components or making changes within a
component. The amount of time teachers had for math instruction was limited to a certain
amount each day, and they said they did not have enough time to do everything. With the limited
time available, teachers are having to prioritize certain curricular components.
Finding 2: Teacher Knowledge Gaps
Based on the need to prioritize due to time constraints, some of the modifications
teachers made were influenced by their knowledge gaps. Subject matter knowledge, pedagogical
knowledge, and pedagogical content knowledge were all factors that influenced what
components of the curriculum were modified. Some teachers reported gaps in subject matter

65
knowledge, which is curriculum knowledge, involving awareness of how topics are arranged in
the school year and over longer periods and ways of using curriculum resources such as
textbooks to organize a program of study for students. Teachers also spoke of gaps in
pedagogical knowledge and pedagogical content knowledge (knowledge of student
understanding, curriculum, and instructional strategies), as well as gaps in knowledge of content.
(Shulman, 1986; Grossman, 1990)
Finding 3: Teacher Self-Efficacy
Teacher knowledge impacted teacher self-efficacy and also played a determining factor
in what was or was not taught in the curriculum. Self-efficacy was even more of an issue due to
the gap between traditional mathematics and Common Core mathematics. As teachers were less
confident in the content knowledge, they were more likely to revert to the math that they were
more confident in teaching, leaving out important components or nuances, because of their
perceived expectation for a successful or unsuccessful lesson. In addition, expectations on
different structures such as collaborative work and being able to manage behaviors deterred
teachers from using specific pedagogical strategies. Both content and delivery of content were
impacted by self-efficacy.
Finding 4: Student Knowledge Gaps
Teachers cited student knowledge gaps as one of the reasons for making modifications
and supplementing the curriculum. Knowledge gaps teachers identified were gaps in fluency,
foundational skills, and word problems. There was a concern about students not having the
prerequisite knowledge or foundational skills needed for the lessons, which prompted them to
make revisions to the curriculum. Some respondents stated they took out the application

66
problems because it was too difficult for their students since they historically struggled with
word problems.
Finding 5: Teacher Ideology

Teacher ideology also played a factor in changes made to the curriculum. Teachers had
different beliefs about the purpose of education, and that drove some of their decisions to modify
the curriculum based on what they believed to be important and whether or not the curriculum
included it. Teacher ideology had an impact on both the content as well as the other components
that go hand in hand with carrying out the content, such as expectations, classroom culture,
engagement strategies, structures, and approach. For example, a respondent shared making
mistakes, joy factor, and collaboration as being elements they valued within their culturally
responsive approach.
Limitations and Delimitations
A limitation of this qualitative study is that I am the primary instrument of the study. As
the primary instrument, I may misinterpret ambiguities of specific words that have multiple
meanings, which can lead to misinterpretation of data collected during the interviews. Although I
have specifically chosen to interview teachers at a different school site from my own due to the
hierarchy of roles, I was acquainted with teachers I interviewed, which may have influenced their
interview responses.
The criteria for the purposive, convenience sample is a TK-6th grade math teacher at
Corner Street Elementary school using the Eureka Math curriculum. A delimitation of the study
is the small sample size with voluntary participation from one school site, which does not allow
for generalizability and is a threat to external validity. In addition, due to the small number, there
may be insufficient statistical power, which could entail low reliability.

67
Implications for Practice
Teachers are underprepared to meet the needs of students in elementary math, especially
during this new era of Common Core mathematics. They are faced with many barriers that can
be difficult to navigate around, which include time, teacher knowledge gaps, student knowledge
gap, teacher self-efficacy, and teacher ideology. This research study shows that changes in
curriculum alone will not improve the current state of math education because the curriculum is
interpreted and delivered in various ways, with the teacher being the medium between the
curriculum and student.
Through interviews, we have learned that teachers modify the curriculum to emphasize,
deemphasize, and supplement different parts of the curriculum. Although some of these
modifications are made by informed decisions based on core practices or students’ needs, many
decisions are driven by limited time, a lack of content knowledge, and a lack of pedagogical
knowledge. The lack of content knowledge is even more pronounced with Common Core math,
which is a new way of teaching and thinking about math. Decisions are also influenced by
teachers’ self-efficacy, teacher ideology, and teacher assumptions. The modifications result in
instruction that is different from the original intent of the curriculum writers. These
modifications have big implications on the student experience and impact on learning while
teachers do not fully understand the impact of some of their decisions.
The current preservice and in-service education and support for teachers is inadequate to
address the needs of teachers and students in providing a focused, coherent, and rigorous math
curriculum and instruction that will increase math achievement in our country, specifically in the
elementary context and in underserved communities. There is a disconnect between theory and
practice as well as between curriculum writers and teachers. This leads to a disconnect between

68
the intent of the curriculum and how that curriculum is implemented. Within the instructional
core, the curriculum only takes into account the content and instructional core, without taking
into consideration the teacher and students.
Akkus (2016) stated that less than half of elementary teachers felt well prepared for
teaching math topics of the Common Core at their grade level as compared to 60% middle school
and 70% high school. Based on the research, there are major gaps in math knowledge at the
elementary level, which negatively impacts instruction and student learning. If teachers do not
fully understand the content they are teaching, they will not be successful in teaching their
students. There is a need for more robust teacher education both at the preservice and in-service
levels. At the preservice level, there needs to be a focus on both the content as well as core
practices in math instruction. According to Zeichner (2005),
teacher educators need to know how to scaffold learning for new teachers and reexamine
and deepen their understanding and ideas that they bring to their preparation programs
and learn how to plan instruction, develop appropriate curriculum, choose instructional
practices and classroom structures, analyze student learning, etc. with diverse groups of
students in constantly changing and uncertain environments. (p. 119)

Ball and Forzani (2009) “referred to teaching as unnatural work that enables others to
learn, understand, think, and do using a flexible repertoire of high leverage strategies and
techniques deployed with good judgement in different contexts”. Teacher educators need to
develop settings in which a practice can be tried out, corrected, refined, and mastered (Lambert
& Graziani, 2009), both in the preservice and in-service setting.
An important pedagogical skill for teachers is orchestrating classroom discussions, which
require teachers to know how to ask questions and pose problems, monitor student participation,
and respond to student ideas (Grossman et al., 2009). Pedagogies of enactment, which are
approximations of practices, allow preservice teachers to rehearse and enact discrete components

69
of complex practice in settings of reduced complexity (Grossman et al., 2009). This type of
support should continue for teachers in service.
Loughran and Berry (2005) discussed the importance of teacher educators explicitly
modeling engaging and innovative teaching procedures while offering students access to the
reasoning that accompanies the practicese. Similar to this, Loughran (2006) discussed the value
of think-aloud and journal writing in making the tacit explicit. Teaching students the “why”
behind the “what” is a critical element of teacher education. If students understand the why
behind what they are doing, they are more adept at applying their learning in different contexts.
Loughran (2006) noted the importance of teacher educators understanding the
complexities of teaching, learning, theory, and practice. Teacher education should include
theoretical and practical work. Unfortunately, there is a separation between theory and university
coursework, on the one hand, and fieldwork in K–12 schools, on the other hand (Grossman et al.,
2009). Grossman et al. (2009) stated that “Teacher education should move from a curriculum
focused on what teachers need to know to a curriculum organized around core practices, in
which knowledge, skill, and professional identity are developed in the process of learning to
practice” (p. 274).
At the in-service level, there needs to be ongoing professional development, support,
feedback, and collaboration opportunities in math curriculum and instruction. At both the
preservice and in-service levels, it is important to create positive experiences in teaching math,
which will help teachers with self-efficacy. There should also be opportunities for teachers to
engage in teachbacks or demonstration lessons with support and feedback. Novice teachers need
many opportunities to practice the instructional routines of core practices. Along with practice,
novice teachers need coaching and feedback with opportunities to apply their feedback. Lampert

70
et al. (2013) discussed the use of rehearsals and deliberate practice in which new teachers try out
routines in simulated settings where the simulated classroom “acts back” in specific ways.
Research has found that teachers’ past experiences with mathematics and perceptions of
teaching effectiveness were associated with mathematics teacher efficacy (Swars, 2005).
Teachers’ mathematical knowledge is central to how well the teacher can use curriculum
materials, assess student progress, and judge how to present, emphasize, and sequence the
material. Preservice teachers need positive experiences within mathematical methods courses to
build efficacious towards teaching mathematics (Swars, 2005). Teachers need positive
experiences within mathematical methods courses towards teaching math to increase self-
efficacy (Swars, 2005).
Research has demonstrated that there are inequalities in the opportunities for students to
learn challenging content, varying across schools, districts, and states, disproportionately
impacting low socioeconomic status communities (Akkus, 2016; Chen & Wang-Ting, 2009).
When students are never exposed to a topic, then it is not possible to learn it. The inequality of
opportunity in learning is a major issue. The research states that access and opportunities for
students, specifically underrepresented students, is an equity issue that stems from teacher
ideology, among others. The teacher is the avenue through which content is delivered to
students, and teachers control what content students are exposed to. For this reason, teacher
implicit bias training should be considered as part of teacher education to create an awareness of
assumptions teachers have about students. This is particularly critical in urban communities
where teachers bring deficit views about students of color (Franke, 2015).
Teacher education should examine teacher ideology and engage teachers in implicit bias
training and critical reflection. Jay and Johnson (2010) discussed the practice of reflection as a

71
critical element of teacher education and progress. They describe reflections through three
different dimensions, which include descriptive, comparative, and critical. Through seeing things
from different perspectives, reflection has the possibility for transformation.
In addition to creating a more robust preservice teacher education experience that
addresses content knowledge, pedagogical knowledge, teacher ideology, and equity, continued
in-service support in these areas is critical to teacher and student success. In-service teachers
should receive continued professional development and coaching on content, core practices, and
curriculum implementation and be provided with opportunities to collaborate with colleagues.

Future Research
To address the gap between curriculum and instruction, more studies examining the
instructional core (teacher, student, content, and task) would provide more insight into the many
variables impacting implementation of a curriculum. Studies examining the relationships
between these elements and how they are interrelated would further inform teacher education
and benefit the field.
These studies will seek to understand how a curriculum is interpreted by the teacher
based on the teachers’ knowledge (content and pedagogical), ideology, self-efficacy, and
perception of students, then delivered to students. These studies will also seek to understand
what type of learning experiences are provided to students based on the teachers’ interpretation
of the curriculum and its impact on student achievement. These findings will have implications
that can be addressed through preservice and in-service teacher education. No amount of
curriculum reform alone will make an impact on student achievement if additional supports are

72
not put into place to support teachers to provide equitable and rigorous learning experiences for
their students.
Potential future research topics to extend this research include:
● Researching differences in responses based on years of experience (i.e., veteran
teachers versus newer teachers).
● Researching differences in how these participants respond to the interview guide in
other disciplines.
● Research on the effectiveness of building elementary math teachers’ content
knowledge in teacher education programs. Math instruction has fundamentally
changed with the implementation of Common Core math standards. Current research
suggests that elementary teachers in the US have less content knowledge than
secondary teachers or elementary teachers in other countries. Therefore, iff teachers
become “experts” in the content they are teaching, would student achievement
increase? And should content competency be a critical component of teacher
education programs?
● Research on the impact of teacher ideology in elementary math instruction
specifically in urban settings. How does teacher ideology affect how teachers modify
curriculum? Are there trends of teacher ideology across different settings? How do
teacher education programs promote critical reflection on matters of equity, and does
it make an impact on student achievement?
● Research on the the core practices (pedagogy) in Common Core elementary math
that support student learning across all contexts.

73
Studies examining the teacher’s role in the implementation of the curriculum across
different contexts serving different communities and needs is critical to address math
achievement. Curricular theorists identify what is critical and how to modify curriculum in
informed ways. Continued research is needed on the different knowledge types, whether content
or pedagogical knowledge, that teachers require and how preservice and in-service educators and
systems can best provide support.
Table 5

Findings Summary
Research Question 2: What are the dominant
reasons teachers modify certain curricular
components of the Eureka Math for their class?
Research Question 1: What components of Eureka
Math do teachers emphasize or deemphasize? How do
they remove, adapt, or supplement parts of the
curriculum?

Time
Teacher Knowledge Gaps
Teacher Self-Efficacy
Student Knowledge Gaps (Perceived)
Teacher Ideology
Emphasized Concept Development
Problem Set
Exit Ticket
Deemphasized (removed
or adapted)
Fluency
Application
Student Debrief
Supplemented Engagement Strategies
(i.e., Collaboration
Structure)
Vocabulary/ Language
Unfinished Learning
Real World Application
Opportunities

Conclusion
Implementation of new curricula is often an approach to address the achievement levels
in math instruction. There is a tendency for school districts to regulate mathematics teaching
practices by mandating the use of a single curriculum as a form of regulation in response to the
failure of schools to raise student achievement levels particularly for students of color and from
low-income communities. In the 1950s–60s, the curriculum was “fixed” and “teacher-proofed”;
teachers were seen as a conduit for curriculum rather than a user or designer. The legacy of this

74
approach continues today, as the teaching profession is under professionalized while teaching is
oversimplified.
Current curricula emphasize mathematical thinking and reasoning, conceptual
understanding and problem solving, and require teachers to play a substantially different role in
the mathematics classroom than has been typical in the US. Teachers bring their knowledge,
beliefs, and experiences to the curriculum, often unintentionally or intentionally diverging from
the intention of the curriculum writers. Elmore (2009) stated that there are only three ways to
improve student learning at scale: raising the level of the content, increasing the skill and
knowledge of teachers, and increasing the level of students’ active learning. In addition, if we
change any single element of the instructional cores (student, teacher, content), we also need to
change the other two (Elmore, 2009). This suggests simply making changes to the curriculum
will have no significant impact on student learning.
There are many potential contributing factors to the current state of mathematics,
including the curriculum being used in the classroom as well as the instruction being provided.
This study has examined the intersection between the curriculum, teacher, and instruction in the
context of elementary math education. It has examined the design of a math curriculum for
essential curricular elements as defined by notable theorists. Furthermore, it has attempted to
identify teacher perception of the curriculum as well as the dominant reasons why and how
teachers modify specific curricular components of the curriculum. The study has sought to
understand the instructional decisions of the teacher based on the curriculum provided, which
ultimately impacts student learning. Based on interview responses, teachers alter the curriculum
in various ways due to a variety of factors that ultimately impacts the learning experience
students receive during the implementation of the curriculum. Teachers emphasized,

75
deemphasized and supplemented parts of the curriculum due to environmental factors, teacher
knowledge, teacher self-efficacy, and teacher ideology. Based on these findings,
recommendations on how to support teachers in pre-service or in-service contexts to improve
math instruction and instruction have been provided.

76
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Abstract (if available)

Abstract This study seeks to understand the role of curriculum in student learning by applying the instructional core framework, which states that there are only three ways to improve student learning at scale: raising the level of the content, increasing the skill and knowledge of teachers, and increasing the level of students’ active learning. The purpose of this study is to examine the curricular components of Eureka Math, a Common Core-aligned math curriculum, to understand what and why components are emphasized, adapted, removed, or supplemented by teachers. Using a semi-structured interview guide, the researcher chose a non-probability, criterion-based, purposeful sampling of seven partipants, who were interviewed to gather qualitative data. The data was transcribed, coded, then organized into themes. Analyses revealed teachers emphasized, deemphasized, and supplemented parts of the curriculum due to environmental factors, teacher knowledge, teacher self-efficacy, and teacher ideology. Findings from the study suggest teacher knowledge gaps in content and pedagogy, teacher ideology, and teacher self-efficacy should be addressed further in both pre-service and in-service teacher education to support student learning. Further research on these topics is also recommended.

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Asset Metadata

Creator Colton, Christin Hwang (author)

Core Title The intersection of curriculum, teacher, and instruction and its implications for student performance

School Rossier School of Education

Degree Doctor of Education

Degree Program Educational Leadership

Degree Conferral Date 2022-05

Publication Date 01/28/2022

Defense Date 11/16/2021

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

Tag math curriculum,math instruction,OAI-PMH Harvest,student achievement

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Kaplan, Sandra (committee chair), Hasan, Angela (committee member), Mora-Flores, Eugenia (committee member)

Creator Email christinshwang@gmail.com,hwangcs@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC110582763

Unique identifier UC110582763

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Format application/pdf (imt)

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Source 20220201-usctheses-batch-910 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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