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The enactment of equitable mathematics teaching practices: an adapted gap analysis

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The enactment of equitable mathematics teaching practices: an adapted gap analysis

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The Enactment of Equitable Mathematics Teaching Practices: An Adapted Gap Analysis
by
Gynelle DeAun Gaskell

Rossier School of Education
University of Southern California
A dissertation submitted to the faculty
in partial fulfillment of the requirements for the degree of
Doctor of Education

August 2022

© Copyright by Gynelle DeAun Gaskell
All Rights Reserved

The Committee for Gynelle DeAun Gaskell certifies the approval of this Dissertation

Yasemin Copur-Gencturk
Darline Robles Committee Co-Chair
Lawrence Picus Committee Chair

Rossier School of Education
University of Southern California
2022

iv

Abstract
Ensuring that all children have opportunities to engage in deep mathematical learning is essential
for our future. Children must learn to value, appreciate, and use mathematics, and see themselves
as knowers and doers of mathematics. Teachers are the key to making this happen and to be
successful, many things need to be in place. The purpose of this study was to understand the
knowledge, motivation, and organizational influences that support teachers’ ability to enact
equitable mathematics teaching practices (EMTPs) that support equitable opportunities for all in
mathematics. Literature revealed several assumed knowledge, motivation, and organizational
influences related to the enactment of EMTPs. Knowledge factors included knowing and
understanding the goals of mathematics and math teaching, knowing and understanding teaching
practices that support equity in the mathematics classroom, knowing and understanding content
knowledge and progressions, as well as knowing how to enact EMTPs. Teacher efficacy, as well
as collective efficacy, were revealed as motivation factors. Identified organizational influences
included having a common vision and philosophy, providing ongoing professional learning, and
having a standards-based curricular resource aligned to the philosophy. Through an adapted gap
analysis based on Clark and Estes's (2008) gap analysis framework, both needs, and assets, were
revealed. Six classroom teachers in Grades 3–5, all of whom had been teaching at the school for
at least 3 years, took part in this qualitative study. Interviews and observations revealed that
teachers have high teacher efficacy for impacting their students’ learning; however, collective
efficacy, as well as all knowledge and organization influences, resulted in needs.
Recommendations, as well as implementation and evaluation plans, are shared in the final
chapters of this dissertation.

v
Dedication
To my mother, who is here in spirit, watching me from Heaven. You taught me to be strong, to
fight for what I want, and to pursue my dreams no matter what. You taught me that adversity can
be overcome and add to your growth. You taught me that studying, working hard, and learning
are important to be the best you can be. I wish you were here so I could show you how hard I’ve
worked, what I have learned, and what I have accomplished. I know you are here though,
watching over me. And I know you would be proud. I was able to accomplish this because you
are a part of me.

To my first math coach, Barb Tanguay. Throughout my career, there have been many people
who have impacted me, and your constant encouragement to ‘be the best math teacher I could
be’ has stayed with me all these years. “If children can say, ‘Tyrannosaurus Rex’, they can say
rhombus” will never be forgotten. You taught me that all students, no matter what, can do hard
things and that I can help them to do it. Thank you for never giving up on me during my first
years of teaching at Altura, and for always making me believe that I could do whatever I set out
to do. Without you, I may not have pursued my career in mathematics teaching.

vi
Acknowledgments
This study could not have been made a reality without the support of many people. Most
importantly, I could not have been successful without the support of my family. To my husband,
John, who encouraged me to pursue this endeavor and supported our family while I was hard at
work. Thank you for pushing me to keep going, for helping me believe that I could do it, and for
endlessly reading my papers to give me feedback. Thank you to my children, Elly and Jacob,
who ensured we stayed a tight-knit family. I was so afraid that we would lose our connection
while I was away studying. Your constant pop-ins to the office to say “hi” and to give me a hug
and kiss or to have a quick chat helped me to keep going. I could feel the love and am forever
grateful to you. And to my other daughter, Shannon, while you were not here, I could feel the
love and encouragement from you. Thank you for always asking how things were going. To my
sister, who listened and encouraged me, and served as “mom.” Thank you. I appreciate you
constantly telling me how proud you are like mom would have done. And to Hank and Gigi, my
beloved French Bulldogs, who kept me company in the office and who were constantly snoring
by my side. Thank you for being wonderful!
I must also acknowledge the amazing teachers I had the opportunity to work with during
this study. Your dedication and passion to support all children is contagious. I am grateful to
have learned so much from you. I appreciate your willingness to share your candid thoughts and
open your classroom for the sake of learning. The organization is lucky to have such
extraordinary people!
To the USC Class of 2022 cohort who supported me and pushed my thinking and skills to
new levels. It was a privilege to come to know you and your unique perspectives. I look forward
to seeing where this work takes all of you! I am especially grateful for the amazing and life-long

vii
friendships I have developed as a result of these past 3 years. You know who you are! Not only
did you support my learning and leadership development, but you fed my soul with love. I feel
so lucky.
Finally, I cannot say how much I appreciate my dissertation chair, Dr. Larry Picus. Thank
you for your encouragement, your knowledge and insights, and your flexibility to help me to be
who I am and communicate it through this work. I could not have done this without you!

viii
Table of Contents
Abstract .......................................................................................................................................... iv
Dedication ....................................................................................................................................... v
Acknowledgments .......................................................................................................................... vi
List of Tables ................................................................................................................................. xi
List of Figures ............................................................................................................................... xii
List of Abbreviations ................................................................................................................... xiii
Chapter One: Introduction .............................................................................................................. 1
Background of the Problem ................................................................................................ 2
Importance of Addressing the Problem .............................................................................. 3
Organizational Context and Mission .................................................................................. 4
Organizational Performance Status ..................................................................................... 5
Organizational Performance Goal ....................................................................................... 6
Description of Stakeholder Groups ..................................................................................... 7
Stakeholder Group for the Study ........................................................................................ 9
Purpose of the Project and Questions ............................................................................... 10
Conceptual and Methodological Framework .................................................................... 11
Definitions ......................................................................................................................... 11
Organization of the Project ............................................................................................... 13
Chapter Two: Review of the Literature ........................................................................................ 14
Influences on the Problem of Practice .............................................................................. 15
Role of Stakeholder Group of Focus ................................................................................ 20
Clark and Estes’ (2008) Knowledge, Motivation, and Organizational Influences
Framework ........................................................................................................................ 21
Stakeholder Knowledge, Motivation, and Organizational Influences .............................. 22

ix
Conceptual Framework: The Interaction of Stakeholders’ Knowledge and
Motivation and the Organizational Context ...................................................................... 49
Summary ........................................................................................................................... 52
Chapter Three: Methodology ........................................................................................................ 54
Conceptual and Methodological Framework .................................................................... 54
Assessment of Performance Influences ............................................................................ 56
Participating Stakeholders and Sample Selection ............................................................. 62
Recruitment ....................................................................................................................... 62
Instrumentation ................................................................................................................. 63
Data Collection ................................................................................................................. 65
Data Analysis .................................................................................................................... 68
Trustworthiness of Data .................................................................................................... 68
Limitations and Delimitations ........................................................................................... 70
Ethics ................................................................................................................................. 72
Positionality ...................................................................................................................... 74
Chapter Four: Results and Findings .............................................................................................. 78
Participating Stakeholders ................................................................................................ 79
Determination of Assets and Needs .................................................................................. 79
Results and Findings for Knowledge Causes .................................................................... 81
Results and Findings for Motivation Causes .................................................................... 95
Results and Findings for Organization Causes ................................................................. 98
Summary of Validated Influences .................................................................................. 103
Chapter Five: Recommendations and Evaluation ....................................................................... 108
Recommendations to Address Knowledge, Motivation, and Organization
Influences ........................................................................................................................ 109
Summary of Knowledge, Motivation, and Organization Recommendations ................. 127

x
Integrated Implementation and Evaluation Plan ............................................................. 130
Limitations and Delimitations ......................................................................................... 146
Recommendations for Future Research .......................................................................... 147
Conclusion ...................................................................................................................... 148
References ................................................................................................................................... 150
Figures......................................................................................................................................... 164
Appendix A: Interview Protocol for Teachers ............................................................................ 165
Appendix B: Observation Tool ................................................................................................... 171
Appendix C: Informed Consent/Information Sheet .................................................................... 176
Appendix D: Recruitment Letter ................................................................................................ 178

xi
List of Tables
Table 1: Organizational Mission, Global Goal and 9
Stakeholder Performance Goals
Table 2: Summary of Assumed Knowledge Influences 37
on a Grade 3–5 Teacher’s Ability to Enact EMTPs
Table 3: Summary of Assumed Motivation Influences 40
on a Grade 3–5 Teacher’s Ability to Enact EMTPs
Table 4: Summary of Assumed Organizational Influences 48
on a Grade 3–5 Teacher’s Ability to Enact EMTPs
Table 5: Summary of Knowledge Influences and Method of Assessment 57
Table 6: Summary of Motivation Influences and Method of Assessment 60
Table 7: Summary of Organization Influences and Method of Assessment 61
Table 8: Knowledge Assets or Needs as Determined by the Data 104
Table 9: Motivation Assets or Needs as Determined by the Data 105
Table 10: Knowledge Assets or Needs as Determined by the Data 106
Table 11: Summary of Knowledge Influences and Recommendations 110
Table 12: Summary of Motivation Influences and Recommendations 120
Table 13: Summary of Organization Influences and Recommendations 123
Table 14: Outcomes, Metrics, and Methods for Internal Outcomes 133
Table 15: Critical Behaviors, Metrics, Methods, and Timing 133
for Evaluation
Table 16: Required Drivers to Support Critical Behaviors 137
Table 17: Evaluation of the Components of Learning for the Program 141
Table 18: Components to Measure Reactions to the Program 143
Appendix B: Observation Tool 171

xii
List of Figures
Figure 1: The Mathematics Teaching Framework 164
Figure 2: Equitable Mathematics Teaching Practices 28
Figure 3: Conceptual Framework: Equitable Opportunities for All 49
Children to Engage in Deep Mathematics Learning
Figure 4: Gap Analysis Overview 55

xiii
List of Abbreviations
CRP Culturally Relevant Pedagogy
CRT Culturally Responsive Teaching
EMTPs Equitable Mathematics Teaching Practices
NCSM National Council of Supervisors of Mathematics
NCTM National Council of Teachers of Mathematics

1
Chapter One: Introduction
The United States has been engaged in mathematics reform for several years to ensure all
children have opportunities for challenging and engaging mathematics learning experiences.
Turner (2016) defined the mathematics reform as “an endeavor to move away from the
traditional, direct instruction approach of the teacher as the sole provider of information toward
the teacher as a facilitator of knowledge” (abstract). Research supports this move towards
equitable teaching practices in which learners engage in mathematical sense-making through
authentic problems (Szabo et al., 2020) and positions children as knowers and doers of
mathematics (Suh et al., 2021). Despite this, inequitable and narrow mathematics experiences
persist in schools, especially for marginalized populations such as those identified as Black,
Latinx, Indigenous, language learners, poor, and with disabilities, among other marginalized
learners (National Council of Teachers of Mathematics [NCTM], 2020).
These inequitable classrooms which utilize traditional practices are characterized by the
teacher standing in the front of the room, demonstrating the knowledge that children “should”
know, followed by the children practicing the demonstrated procedure (Boaler, 2008; Mokros et
al., 1995; Seeley, 2009; Turner, 2016). Teacher questioning is predominantly closed, seeking
one- or two-word answers, usually asking for an answer to the problem. This type of teaching
enforces the message that mathematics is a body of knowledge to be learned linearly, that it is
about rules and procedures and one way to do something, and that the teacher holds the
knowledge, and therefore the power.
Not surprisingly, this causes an opportunity gap, seriously impacting many children,
especially those who are traditionally marginalized such as children of color, English language
learners, and those who struggle (National Council of Teachers of Mathematics, 2020). The most

2
recent National Assessment of Education Progress (NAEP) (2019) scores for mathematics
indicate 41% and 34% proficiency (or above) rates for fourth and eighth graders, respectively,
which have changed very little since 2007. Even more disturbing are the race discrepancies. Of
fourth grade children across the nation, 52% of White children are proficient or higher as
compared to 20% Black and 28% Hispanic. Asian/Pacific Islanders and Asians outperformed
their White peers by approximately 15 percentage points. The trend was the same across eighth-
grade children: White 44%, Black 14%, Hispanic 20%, Asian/Pacific Islander 62%, and Asian
64% (NAEP, 2019).
Background of the Problem
Mathematics has a history of being inequitable. In the United States, mathematics was
initially intended to serve privileged, White males (Broussard & Joseph, 1998; California
Department of Education et al., 2021). During the early 1900s, an influx of immigrants arrived
on the shores of the United States. At the time, Social Darwinists believed that those from lower
classes were inferior, resulting in the idea that some children had less innate ability than others
(Crosby & Owens, 1993). As such, children were sorted and classified based on perceived ability
level. Within these tracks, it was suggested that different types of instruction should be delivered
to prepare different children for their “rightful stations in life” (Loveless, 1998, p. 2). The lower
tracks often resulted in passive instruction and rote learning where children replicated the
thinking of the teacher (Oakes, 1992), resulting in a narrow understanding of mathematics for
many children, especially those traditionally marginalized populations, including by not limited
to people of color, those of lower socioeconomic status, those with learning disabilities, and
those who are multilingual. This effectively denies access to opportunities for deep mathematical
learning for all. Despite many attempts to reform mathematics teaching practices to provide

3
equitable opportunities for engaging in deep mathematics learning for all, the practice of sorting
and classifying based on beliefs about ability, whether conscious or unconscious, continues.
Additionally, it has been found that even teachers who believe in mathematics teaching
approaches that provide access for all to engage in deep mathematics learning, can consciously
or unconsciously revert to traditional approaches that they experienced, and which have been
integrated into their teaching practices (Wright, 2012). Bourdieu and Wacquant (1992) referred
to this socialized subjectivity as “habitus” (p. 126).
From the late 1950s until today, the United States has been engaged in “Math Wars”
(Wright, 2012). Progressives have argued for a focus on problem-solving and deep mathematics
learning for all (Wright, 2012) while others argued for a more back-to-basics curriculum
(National Research Council and Mathematics Learning Study Committee, 2001; Seeley, 2009).
Despite the work of the organizations such as the National Council of Teachers of Mathematics
(NCTM) which aim to ensure equity of opportunities for deep mathematics learning for all,
inequities still exist in our classrooms, especially for marginalized populations. These
populations include but are not limited to, people of color, those of lower socioeconomic status,
those with learning disabilities, and those who are multilingual.
Importance of Addressing the Problem
The importance of ensuring that all children have access to deep mathematics learning is
important for many reasons. To begin, mathematics provides opportunities for success in life
(English & Gainsburg, 2015) Unfortunately, mathematics in its current state serves as a
gatekeeper, distributing opportunities to those who are deemed successful (Burdman, 2018).
Unfortunately, the way mathematics is taught in some classrooms, the way assessments are used

4
to sort and track children, and the ideas about what success is in mathematics and who can be
successful in mathematics prevent some children from realizing opportunities.
To be prepared for the 21st century, children need to be able to apply skills to solve
unknown problems (English & Gainsburg, 2015; Gasser, 2011). In traditional classrooms,
children are not engaged in problem-solving that requires the use of 21st-century skills such as
collaboration, communication, creativity, and critical thinking. As such, they will not have
access to many opportunities as they will most likely not be prepared for the future workforce or
their lives (Suh et al., 2021; Toheri et al., 2020).
Furthermore, engaging in equitable mathematics teaching practices (EMTPs) positions
children as competent knowers and doers of mathematics (Suh, et al., 2021), promoting positive
mathematics identities for all children. Aguirre et al., (2013) defined a mathematical identity as
“the dispositions and deeply held beliefs that children develop about their ability to participate
and perform effectively in mathematical contexts and to use mathematics in powerful ways
across the contexts of their lives” (p. 14). When mathematics is viewed through a narrow lens in
which there is only one way to get an answer and where the teacher holds the power, children
feel excluded from the mathematics, oftentimes limiting their opportunities to develop positive
mathematics identities (Louie, 2017). This ultimately results in poor performance and
withdrawal from mathematics.
Organizational Context and Mission
Green View River International School (GVRIS), a pseudonym, is a private, non-profit
pre–Kindergarten to Grade 12 international school located in Southeast Asia, whose vision is to
develop exceptional citizens prepared for a changing world. The mission is to provide each
student with American education with a global perspective. GVRIS is a large American-

5
curriculum school with an approximate enrollment of 4,000 children for the 2020–2021
academic year. American passport holders represent approximately 50% of the 2020–2021
student body. The remaining children hold passports from over 60 countries, with a large
majority from Asian countries. GVRIS provides support for children with mild to moderate
learning needs and additional support for English as a second language is offered from Grades 1–
3.
During the 2019–2020 school year, GVRIS employed approximately 400 teachers, 81%
of whom hold advanced degrees (5% Doctoral degrees; 76% Master’s degrees). These teachers
are from a variety of countries including approximately 60% from the United States, and the rest
from Asia, Australasia, Canada, Europe, South Africa, and South America.
The most recent strategic plan indicates two areas of focus related to this study:
1. High-quality tier 1 instructional practices including differentiation to support a range
of learners
2. A culture of inclusivity that supports all children feeling valued
Organizational Performance Status
The organizational performance problem at the root of this study was the inconsistent
enactment of EMTPs (Bartell et al., 2017; California Department of Education et al., 2021;
NCTM, 2020) resulting in inequitable opportunities for all children in the elementary school to
engage in deep mathematics learning. During the most recent math review, learning walks
revealed that some children were being taught in traditional, unidimensional classrooms where
children are expected to copy what the teacher is doing without much understanding of why,
while in other classrooms, children were engaged in collaborative problem-solving, which
supports deep conceptual understanding and application of mathematical concepts. Additionally,

6
through coaching work done by the instructional coaches in Grades 3–5, similar patterns had
been observed. Furthermore, there was evidence from conversations about mathematics that
teachers had varying perspectives on the purposes of mathematics, and some were unaware of
how their instructional practices may impact children’s opportunities for learning deep
mathematics. Finally, a student survey and student focus groups in the elementary school
revealed inequitable opportunities for engaging in deep mathematics learning as some children
described traditional teaching methods while others described experiences that provided access
to deep mathematics learning for all children. This problem impacts the school’s vision and
mission as they are dedicated to developing exceptional citizens prepared for a changing world
for each student. When there are inequitable learning opportunities for children in mathematics,
not all children are not prepared for the future.
Teachers have a direct impact on student learning. Research suggests that teachers have
two to three times the impact on student achievement than any other school-related factor
(RAND Corporation, 2019). Boaler (2015) also suggested that all students can learn mathematics
at high levels if given the right opportunities. If some teachers are teaching in traditional ways,
they are limiting some children from gaining deep mathematical understanding, thus creating
opportunity gaps (Flores, 2007). As mathematics is a “key mechanism is the distribution of
opportunity,” (Burdman, 2018), children, especially those who are traditionally marginalized
such as children of color, English language learners, and those who struggle, are missing out.
Organizational Performance Goal
As part of GVRIS’s strategic plan, one area of focus is high-quality tier 1 instructional
practices including differentiation to support a range of learners. As a result of this area of focus,
by May 2027, every student will engage in deep mathematics learning experiences with a focus

7
on content and 21st-century skills. This focus area was developed by the curriculum office based
on the engagement of various stakeholder groups over 18 months in both formal and informal
settings such as the faculty coffees and community sessions. GVRIS’s strategic plan was
approved by the school board in March 2021.
As part of this broader focus, GVRIS’s goal is that 100% of mathematics teachers will
enact defined EMTPs by May 2023, resulting in equitable opportunities for all children to
develop deep mathematical learning. This specific performance goal related to the enactment of
EMTPs was developed by the K–12 math team which consisted of a teaching and learning
leader, deputy principals, math department chairs and leaders, as well as instructional coaches.
This goal was based on 12 months of research in the form of a literature review to understand
EMTPs, student, faculty, alumni, and parent focus groups, a student survey, and learning walks.
Approval of the goal was made by the assistant superintendent and the superintendent.
Description of Stakeholder Groups
The elementary classroom teachers at GVRIS are the primary contributing stakeholder
group. These teachers are responsible for the enactment EMTPs resulting in deep mathematics
learning for all children. The mathematics coach as well as elementary instructional and PLC
coaches organize, plan, and facilitate professional development and classroom coaching support
for teachers so that they can effectively enact EMTPs. The purpose of other coaches beyond the
mathematics coach providing professional development is to build capacity beyond that of the
mathematics coach. The mathematics coach trains all instructional and PLC coaches in
mathematics to ensure consistency in professional learning experiences. The assistant principals
oversee the professional development plan and progress of the professional learning experiences
as well as the overall progress of teacher enactment of EMTPs. Furthermore, the PreK–12 math

8
team, led by the teaching and learning leader, creates and communicates PreK–12 mathematics
learning principles aligned to the organization’s mission and vision and based on current
literature on EMTPs. This provides a common language and philosophical approach for the
teachers, instructional coaches, and assistant principals. Finally, the children benefit from the
achievement of the organization’s performance goals. When EMTPs are enacted in all
classrooms, all children will have opportunities to learn mathematics at high levels and develop
positive mathematical identities. Table 1 describes the stakeholders’ performance goals.

9
Table 1
Organizational Mission, Global Goal, and Stakeholder Performance Goals
Organizational mission
Green View River International School is committed to providing each student with American
education with a global perspective.
Organizational performance goal
By May 2027, all students will demonstrate deep mathematics learning of content and
mathematical practices standards, as well as the learning aspirations, as measured by unit
assessments.
Teachers
By May 2023,
elementary
teachers at
GVRIS will
consistently
enact EMTPs
as defined by a
school-
designed
rubric.
Math coach
By December 2023, the
elementary
mathematics coach,
in conjunction with
other instructional
coaches and PLC
coaches, at GVRIS
will provide monthly
professional
development sessions
on EMTPs as defined
by a school-designed
rubric. Session
frequency will be
indicated on the PLC
calendars.
Core math team
By December 2021, the
core math team will
create and communicate
PreK–12 mathematics
learning principles
aligned to the
organization’s mission
and vision and based on
current literature on
EMTPs. This goal will
be met upon successful
communication at a
board meeting, in the
school’s weekly e-letter,
and at division faculty or
PLC meetings.

Assistant principal
By December 2023,
the assistant
principal will
evaluate teachers’
enactment of
EMTPs as defined
by a school-
designed rubric
and measured by
formal
observations and
professional
growth and
development
plans.

Stakeholder Group for the Study
All stakeholders are essential in meeting the organization’s performance goal. However,
elementary classroom teachers are the primary stakeholder group as they are responsible for
enacting EMTPs with children. Therefore, it is important to understand their perspectives on
knowledge, motivation, and how organizational influences both support and inhibit their ability
to enact the EMTPs. The stakeholder’s goal, supported by the Instructional and PLC coaches and

10
assistant principals, is that 100% of elementary classroom teachers will enact EMTPs in their
classrooms as defined by a school-designed rubric. This goal was developed by the K–12 math
team which consists of a teaching and learning leader, assistant principals, math department
chairs and leaders, as well as instructional coaches. This goal was based on 12 months of
research in the form of a literature review to understand EMTPs, student, faculty, alumni, and
parent focus groups, a student survey, and learning walks. Approval of the goal was made by the
assistant superintendent and the superintendent. Progress will be tracked through formal and
informal observations in collaboration with the instructional coach, administrators, and math
leaders. Additionally, progress will be measured by assistant principals through evaluation plans.
Purpose of the Project and Questions
The purpose of this study was to conduct an adapted gap analysis to examine the
knowledge, motivational, and organizational influences impacting the successful enactment of
equitable mathematics practices so that all elementary children are afforded opportunities to
engage in deep mathematics learning and to develop positive mathematics identities. While a
complete gap analysis would have focused on leaders and children, for practical purposes,
elementary mathematics teachers were the focus of this analysis. The analysis began by
generating a list of possible or assumed influences that were examined systematically to focus on
actual or validated causes.
As such, the questions that guided this study were the following:
1. What is the current status of teachers’ knowledge and motivation related to enacting
equitable mathematics teaching practices (EMTPs), or those teaching practices that
provide access to deep mathematics learning for all learners?

11
2. How do organizational factors influence teachers’ capacity to enact equitable
mathematics teaching practices (EMTPs)?
Conceptual and Methodological Framework
A gap analysis is a systematic, analytical inquiry method that helps to clarify
organizational goals and determine the causes of performance gaps in relation to knowledge,
motivation, and organizational processes (Clark & Estes, 2008; Rueda, 2011) and is one system
for improving human performance. In this study, The Clark and Estes framework was adapted as
an improvement study to examine the knowledge, motivation, and organizational factors that
influence the organization’s goal. As I sought to deeply understand and describe the knowledge,
motivation, and organizational influences, this study engaged in a qualitative methodological
approach utilizing semi-structured interviews and classroom observations. Current literature and
personal knowledge informed the assumed knowledge and skills, motivation, and organizational
influences impacting the organizational performance goal. Research-based solutions will be
recommended and evaluated comprehensively.
Definitions
• Adaptive reasoning: The “capacity for logical thought, reflection, explanation, and
justification” (National Research Council and Mathematics Learning Study
Committee, 2001, p. 116)
• Conceptual understanding: “Comprehension of mathematical concepts, operations,
and relations” (National Research Council and Mathematics Learning Study
Committee, 2001, p. 116)
• Deep mathematics learning: The “development of strands of mathematical
proficiency, including conceptual understanding, procedural fluency, and problem-

12
solving and reasoning” through high cognitive demand tasks that allow for multiple
strategies, solution paths, and representations, and that require students to analyze,
compare, justify, and prove their solutions through discourse (Aguirre & del Rosario
Zavala, 2013, p. 43)
• Equitable mathematics teaching practices (EMTPs): Mathematical teaching practices
that “provide every student with access to meaningful mathematics by leveraging
students’ strengths, drawing on students as sources of knowledge, and challenging
spaces of marginality...and position each and every student to make sense of
mathematics and develop positive mathematics identities” (Huinker & Bill, 2017, p.
6)
• Equity in mathematics: Access to deep mathematics learning for all children that
builds on their cultures, strengths, and experiences, and supports them in seeing
themselves as knowers and doers of mathematics
• Mathematics reform: Turner (2016) described mathematics reform as “an endeavor to
move away from the traditional, direct instruction approach of the teacher as the sole
provider of information toward the teacher as a facilitator of knowledge” (abstract)
• Procedural fluency: The “skill in carrying out procedures flexibly, accurately,
efficiently, and appropriately” (National Research Council and Mathematics Learning
Study Committee, 2001, p. 116)
• Professional growth and development plan: Green View River International School’s
teacher evaluation plan
• Strategic competence: The “ability to formulate, represent, and solve mathematical
problems” (National Research Council, 2001, p. 116)

13
• Traditional mathematics teaching: A passive, unidimensional approach to
mathematics instruction characterized by children’s replication of teacher’s strategies,
silence, and rote memorization where the goal is the transmission of knowledge
(Boaler, 2008; Mokros et al., 1995; Seeley, 2009; Turner, 2016)
Organization of the Project
Five chapters are used to organize this study. This chapter provided the reader with the
key concepts and terminology commonly found in a discussion about EMTPs and teachers’
inconsistent enactment of these practices. The organization’s mission, goals, and stakeholders, as
well as the initial concepts of gap analysis were introduced. Chapter 2 provides a review of the
current literature surrounding the scope of the study and details the knowledge, motivation, and
organizational influences. Topics including goals of mathematics, causes of inequities in
mathematics, as well as practices that constitute equitable practices in mathematics and how to
enact them, will be addressed. Chapter 3 details the methodology when it comes to the choice of
participants, data collection, and analysis. In Chapter 4, the data and results are assessed and
analyzed. Chapter 5 provides solutions, based on data and literature, for closing the perceived
gaps as well as recommendations for an implementation and evaluation plan for the solutions.

14
Chapter Two: Review of the Literature
Inequitable opportunities for all children to engage in deep mathematics learning persist
despite repeated efforts for mathematics reform in the United States. Attempts to move towards
more equitable approaches which position children as knowers and doers of mathematics,
resulting in positive mathematics identities have only been somewhat successful (Rosenquist et
al., 2015; Szabo et al., 2020; Suh et al., 2021). Instead, more traditional approaches that result in
passive or negative mathematical identities and narrow purposes for learning mathematics
remain for many. Unfortunately, marginalized populations such as those identified as Black,
Latinx, Indigenous, English language learners, poor, and with disabilities, among others are more
likely than White children to have instruction that engages in more traditional methods, widening
the opportunity gap even further (NCTM, 2020).
Mathematics is both a gateway and a gatekeeper (Aguirre et al., 2013; Burdman, 2018,
NCTM, 2020). As such, opportunities for engaging in deep learning of mathematics matter. The
21st century requires citizens to be able to apply skills to solve unknown problems (English &
Gainsburg, 2015; Gasser, 2011). In a study conducted with over 300 high school students, Boaler
(2002) found that those in traditional settings often view “school” mathematics and “life”
mathematics as different entities, thus preventing the transfer of knowledge and skills.
Alternatively, Boaler (2002) found that those who engage in deep mathematics learning where
they are positioned as sense-makers were able to solve problems in different situations. For
students who are not provided these deep mathematics learning experiences, opportunities in life,
and to feel as if they are successful in mathematics, are diminished.
In this chapter, I will first review the history and origins of inequitable opportunities for
mathematics as well as how the United States has attempted to close these opportunity gaps for

15
learning deep mathematics for all. Following this, I will review the role of elementary
mathematics teachers in supporting equitable opportunities for all children to engage in deep
learning of mathematics, followed by the explanation of the knowledge, motivation, and
organizational influences, specifically with regard to Grades 3–5 mathematics teachers, used in
this study. Finally, I will complete the chapter by presenting the conceptual framework.
Influences on the Problem of Practice
Historical Perspective
Mathematics in the United States has been inequitable since its inception. Before the
1900s, most schools provided education to middle- or upper-class White males to prepare them
for elite universities (California Department of Education et al., 2021) leading to further
opportunities. The idea of tracking began in the 1930s when an influx of immigrants arrived in
the United States. Those who were considered highly intelligent were placed in tracks designed
to prepare them for post-secondary education or careers while those deemed less intelligent were
placed into tracks characterized by rote, passive learning (Hallinan, 1994).
In 1957, the Soviet Union launched Sputnik, the world’s first space satellite. The United
States panicked, worried that they were losing ground as the world leader. In response, they set
out to create a new generation of scientists and mathematicians through reform. To create more
mathematicians and scientists, various educational groups emerged, creating different courses
and curriculum. These courses and curriculum moved away from more traditional forms of
teaching with a focus on memorization of basics to a focus on problem-solving and deep
mathematics learning for all (Seeley, 2009). Many criticized the newly produced curricula as
lacking focus on basic skills. Additionally, teachers were not equipped to teach such in-depth
content, and eventually, many reverted to a back-to-basics curriculum with a focus on procedures

16
and skills (Wright, 2012). However, more progressive mathematicians did not stand by idly. In
the 1980s, it was recognized that the quality of math and science was deteriorating. Reformers
saw mathematics as a barrier to economic and social advancement and advocated for progressive
approaches to support learning for all (Wright, 2012). In response, the National Council of
Teachers of Mathematics (NCTM) issued a new set of curriculum standards, again with a focus
on learning through problem-solving. As with previous efforts, this was not without dispute. For
example, in 1998, California found the progressive approaches to be insufficient in part due to
lack of training and the higher demands through teaching through a problem-based approach
(Wright, 2012).
The “Math Wars” continue today. From “new math” to Back-to-Basics curriculum
(Seeley, 2009) to policies such as the No Child Left Behind Act (2001) and its updated version,
the Every Student Succeeds Act (2015), Americans continue to search for ways to improve
mathematics outcomes in the United States. In 2010, the Common Core State Standards for
Mathematics were finalized, offering quality standards for all children (National Governors
Association Center for Best Practices, Council of Chief State School Officers, 2010). A key
feature of the Common Core State Standards for Mathematics is the K–12 Standards for
Mathematical Practices, elaborated from the NCTM Process Standards (NCTM, 2000). These
practices define how children should engage in learning mathematics (NCTM, 2020). In 2014,
the National Council of Teachers of Mathematics (NCTM) published Principles to Actions:
Ensuring Mathematical Success for All. This framework provided guidance for strengthening
mathematics teaching to promote deep mathematics learning for all children. The practices
within the framework are offered in conjunction with five guiding principles of school
mathematics programs that must be in place for successful enactment of the teaching practices,

17
one of which is a commitment to access and equity (NCTM, 2014). Despite these attempts,
successful implementation of instruction that promotes understanding, reasoning, and
communication is typically only found in pockets, creating inequitable opportunities to learn
deep mathematics (Rosenquist, 2015).
Equity in Mathematics
Equity in mathematics has been defined by authors and researchers in various, yet mostly
similar ways. Gutierrez (2002) stated that equity in mathematics is the inability “to predict
mathematics achievement and participation based solely on student characteristics such as race,
class, ethnicity, sex, beliefs, and proficiency in the dominant language” (p. 153). NCTM (2014)
suggested that equitable mathematics “requires that all children have access to a high-quality
mathematics curriculum, effective teaching and learning, high expectations, and the support and
resources needed to maximize their learning potential” (p. 59). Further, Felton-Koestler (2019),
developed different views related to equity in mathematics. They are:
• Equity as student-centered mathematics (equity as SCM): SCM as equity because it
allows more children to learn important mathematics
• Equity as accepting all cultures (equity as accepting): Allowing children to use
strategies (especially algorithms) from other countries
• Equity as actively seeking out cultural connections (equity as seeking): Purposely
designing problems that draw on children’s backgrounds and prior experiences
• Equity as teaching math for social justice (equity as TMfSJ): Connecting
mathematics to controversial or ‘‘social justice’’ issues (p. 159)
Aguirre et al. (2013) combined pieces of these definitions and suggested that,

18
all children, in light of their humanity—their personal experiences, backgrounds,
histories, languages, and physical and emotional well-being— must have the opportunity
and support to learn rich mathematics that fosters meaning-making, empowers decision-
making, and critiques, challenges, and transforms inequities and injustices. Equity does
not mean that every student should receive identical instruction. Instead, equity demands
that responsive accommodations be made as needed to promote equitable access,
attainment, and advancement in mathematics education for each student. (p. 9)
While a variety of definitions of equity have been suggested, this study will use the definition of
equity in mathematics as follows: all children have access to deep mathematics learning that
builds on their cultures, strengths, and experiences, and supports them in seeing themselves as
knowers and doers of mathematics.
To best understand how to support equity in mathematics, it is important to understand
the causes of inequities. As such, the next section will address the root causes of inequities in
mathematics.
Causes of Inequities in Mathematics
There are several known reasons why children do not have equitable opportunities for
deep mathematics learning. To begin, broad societal influences continue to perpetuate
opportunity gaps including, but not limited to, disparities in resources and access to strong
curriculum and teachers (Burdman, 2018; Louie, 2017; NCTM, 2020). Additionally, certain
types of mathematical knowledge have been legitimized meaning that the experiences and ways
of knowing of many children are not recognized (Yolcu, 2019). Ernest (1991) suggested that the
standards, assessments, and language in the United States remain “traditionally produced by
White middle-class males and presented in an unapplied and ‘decontextualized’ manner” (p.

19
267). This causes great opportunity gaps, especially for our most marginalized populations.
These populations include but are not limited to, people of color, those of lower socioeconomic
status, those with learning disabilities, and those who are multilingual.
According to NCTM (2020), other structures contributing to opportunity gaps include
assessment practices, tracking and ability grouping, and inflexible implementation of curricular
resources. An overemphasis on high-stakes testing and accountability pressures lends itself to a
particular view of mathematics in which the goal is correctness and to perform (Clark et al.,
2014; Felton-Koestler, 2019). Further exacerbating the situation is testing, which is used to sort,
track, and label children, resulting in severe damage to children (Burdman, 2018). This is
especially true for children of color who tend to be overrepresented in the lower tracks (Morvan,
2017). These labeling systems often perpetuate a deficit mindset (NCTM, 2020). Celadon-
Pattichis et al. (2018) also found that instruction in the lower tracks was less complex, further
decreasing opportunities for learning of deep mathematics for all children and impacting how
children see themselves as knowers and doers of mathematics (Boaler, 2015). Smith and Stein
(2011) reported that those in lower tracks with narrow curriculum have lower achievement in
mathematics.
Yet another cause of inequitable opportunities in mathematics is implementing
curriculum materials with fidelity, in other words, following the resource exactly as written.
When teachers cannot use their professional judgment and what they know about their children’s
needs, interests, experiences, and cultures, children may feel disconnected, and mathematics
becomes less meaningful (NCTM, 2020).
Beliefs about the nature of mathematics, how mathematics should be taught, and who can
be successful in mathematics also perpetuate inequitable opportunities for all children to engage

20
in deep mathematics learning. Some people view mathematics as a collection of discrete skills to
be received (Louie, 2017). When viewed through such a narrow lens in which there is one way to
solve problems, and the purpose is to replicate the skills taught by the teacher (Burdman, 2018;
Louie, 2017), children are excluded from developing positive mathematical identities (NCTM,
2020) and engaging in deep mathematics learning. This often results in low achievement and
disinterest in pursuing further mathematics (Burdman, 2019) and unintentionally excludes those
children who do not think similarly to the teacher. These conditions create what Louie (2017)
termed a “culture of exclusion” (p. 489).
Further is the idea of who is capable of engaging in deep mathematics learning as a way
to exclude certain children from opportunities for deep mathematics learning. Some believe that
mathematics ability is innate and only some children are good at math, thus labeling those who
are not perceived as “gifted” as “low ability” (Louie, 2017). These children are typically viewed
from a deficit perspective (Suh et al., 2021). Further, they tend to receive low-level instruction
and mathematics experiences that consist of rote learning and reinforcement of skills (NCTM,
2020). These inequities inevitably result in negative math identities and poor achievement, thus
perpetuating the cycle (Burdman, 2018).
Role of Stakeholder Group of Focus
Teachers are the main in-school influence on children’s academic achievement (RAND,
2019) and contribute to the development of the identities of children (Burdman, 2018; NCTM,
2020). How teachers interact with children and what messages they give (Mueller & Dweck,
1998) impact how children view the nature of knowledge and learning as well as themselves as
learners (NCTM, 2020). For example, in traditional teacher-centered classrooms where the
teacher is the holder of all knowledge, transmission and performance are the main purposes of

21
learning. Knowledge here is conceptualized as a fixed body of knowledge (Louie, 2017) in
which there are single ways to solve problems (Burdman, 2018) and typically begin with abstract
mathematics with a focus on memorization of procedures to be applied to problems later (Felton-
Koestler, 2019). In these classrooms, oftentimes there is a belief that only some children are
capable of deep mathematics learning (NCTM, 2020). As such, children will begin to view
themselves as incapable of doing math if they do not think in the way of the teacher (Huinker &
Bill, 2017). Alternatively, student-centered classrooms are those that provide opportunities for
children to engage in learning and problem-solving (Felton-Koestler, 2019) and support the
development of positive student identities through positioning them as knowers and doers of
mathematics (Huinker & Bill, 2017). Additionally, student-centered classrooms support the idea
that mathematical knowledge is fluid, flexible, and constructed (Anthony et al., 2015; Wachira &
Mburu, 2017). Ultimately, the decisions teachers make and the types of experiences they provide
impact learning and how children view themselves as mathematicians as well as the nature and
purposes of mathematics (NCTM, 2020). Fortunately, there are practices that teachers can enact
that will support children in positively experiencing mathematics. As such, this chapter will
examine influences related to teachers, specifically Grades 3–5 teachers of mathematics, as they
are key in supporting equitable opportunities for all children to engage in deep mathematics
learning. The following section outlines the framework used to examine the related knowledge,
motivation, and organizational influences.
Clark and Estes’ (2008) Knowledge, Motivation, and Organizational Influences
Framework
The Clark and Estes (2008) framework is a problem-solving process used to study
stakeholder performance within an organization. It provides a way to clarify stakeholder and

22
organizational goals, assess performance or achievement of the goals, and describe the gaps in
stakeholder performance with regards to organizational goals. This study was adapting the
framework to determine the level of knowledge, motivation, and organization (KMO) based on
the literature that follows. In the next section, KMO assumed influences of elementary
mathematics teachers will be addressed.
Stakeholder Knowledge, Motivation, and Organizational Influences
Knowledge and Skills
To meet goals, stakeholders must understand the types of knowledge required to meet
them (Rueda, 2011). Anderson and Krathwohl (2001) highlighted four specific types of
knowledge: factual, conceptual, procedural, and metacognitive. Factual knowledge refers to the
basic, concrete knowledge within a discipline while conceptual knowledge describes more
complex ideas, theories, or mental models. Procedural knowledge refers to the knowledge of
how to do something. Finally, metacognitive knowledge is the awareness of one’s own cognitive
processes (Anderson & Krathwohl, 2001; Rueda, 2011).
Literature suggested different types of knowledge required of teachers to enact equitable
mathematics teaching practices (EMTPs). Each of the identified knowledge types found in the
literature was categorized using Anderson and Krathwohl’s (2001) terms. To begin, teachers
must know and understand the goals of mathematics and mathematics teaching (NCTM, 2020).
Additionally, teachers must know and understand the mathematical content and progressions to
be taught (Hill et al., 2008; Suh et al., 2021). Furthermore, teachers must know and understand
what denotes EMTPs (NCTM, 2017). In terms of procedural knowledge, teachers must know
how to enact the defined EMTPs (NCTM, 2014). Metacognitive knowledge, while important, is
beyond the scope of this study.

23
Goals of Mathematics and Mathematics Teaching
For teachers to ensure all children have access to deep mathematics learning, literature
suggested that they must understand the goals of mathematics and mathematics teaching. For
teachers who view mathematics as a process of transmission of a static body of knowledge,
traditional, passive teaching remains the appropriate approach. Unfortunately, this purpose and
approach excludes children not only from engaging in deep mathematics learning but oftentimes
leads to negative mathematics identities. Additionally, this type of teaching suggests to children
that mathematics is about performing procedures that have been taught by the teacher. Instead,
children should experience mathematics in a way that helps them “develop deep mathematical
understanding as confident and capable learners” (NCTM, 2020, p. 3).
Further, mathematics has beauty and children should engage in the joy and wonder of
mathematics (Boaler, 2008; California Department of Education et al., 2021; NCTM, 2020).
Only when teachers view the subject as one that can be explored and appreciated, can children
see the beauty, wonder, and joy. Experiences in which children can ask questions, see themselves
in the mathematics they are exploring, and create meaning, are those that teachers need to
understand, value, and provide.
Additionally, mathematics should be viewed as a tool to understand, critique, and change
the world (NCTM, 2020). Children must engage in authentic experiences in which they see how
math is connected to their lives, not a subject that is only done in school (English & Gainsburg,
2015). Gutstein (2012) described “reading the world” with mathematics as a way of developing
an understanding of situations and “writing the world” with mathematics as a way of acting on
the knowledge to make positive change. Yolcu (2019) argued that children must use mathematics
to better understand society and prepare to be active citizens in a democratic world. Only when

24
teachers understand that mathematics is more than just a body of knowledge to be memorized,
can they truly provide experiences in which all children can benefit greatly from these important
goals of mathematics.
Equitable Mathematics Teaching Practices
EMTPs were described throughout the literature. One practice that has been found to
promote equitable opportunities for deep mathematics learning is inquiry-based teaching which
employs a constructivist approach (Wachira & Mburu, 2017). Inquiry-based teaching allows
children to construct knowledge of mathematics in ways that make sense to them, which is at the
heart of reform mathematics instruction. NCTM (2020) supports this idea and suggests that
equitable practices should aim for “shared mathematics learning experiences through inquiry-
based activities focused on robust understanding for all children” (p. 27) rather than grouping by
ability. Additionally, inquiry-based teaching is equitable in that it provides opportunities for
children to solve problems in a variety of ways and allows teachers to build on children’s
strengths and experiences (Gutierrez & Irving, 2012; Huinker & Bill, 2017). In turn, children
with a variety of needs are supported (Gutiérrez and Irving, 2012; NAEYC, 2019). Inquiry-based
teaching can also support mathematical sense-making through authentic problem-solving
(Huinker & Bill, 2017; Szabo et al., 2020). This positions children as knowers and doers of
mathematics (Suh et al., 2021) who develop positive mathematics identities (Aguirre et al.,
2013). Further evidence suggests that all children, including those with perceived low ability and
those with disabilities, can gain confidence in mathematics, develop positive identities, and make
gains in mathematics achievement through inquiry-based approaches (Boaler, 2015; Tan et al.,
2019).

25
Classrooms that utilize an inquiry-based approach engage children in a series of
mathematical practices, allowing for participation in, and learning of, mathematics. The skills
learned are typically transferable to other contexts beyond mathematics (National Governors
Association Center for Best Practices, Council of Chief State School Officers, 2010). They
include but are not limited to reasoning and critical thinking, analysis of ideas, inferential
thinking, engaging in discourse, strategically selecting and using tools, proving thinking based
on models, and critiquing the reasoning of others. Through the practices, children have voice and
choice about different approaches and solutions, as well as strategies and tools, which support
agency and release the power to the children, creating opportunities for all children to engage in
deep mathematics learning. Some refer to this type of instruction as “ambitious” (NCTM, 2020;
Windschitl et al., 2011). Anthony et al. (2015) added that ambitious mathematics teaching also
involves teachers being able to skillfully listen and respond to children as they engage in
problem-solving, positioning children as knowledgeable so that they believe that they are
capable mathematicians.
To ensure that children are engaged in such experiences as described above, NCTM
(2014) suggested the use of eight effective mathematics teaching practices that “represent a core
set of high-leverage practices and essential teaching skills necessary to promote deep learning of
mathematics.” These eight mathematics teaching practices are as follows:
• Establish mathematics goals to focus learning
• Implement tasks that promote reasoning and problem-solving
• Use and connect mathematical representations
• Facilitate meaningful mathematical discourse
• Pose purposeful questions

26
• Build procedural fluency from conceptual understanding
• Support productive struggle in learning mathematics
• Elicit and use evidence of student thinking (p. 9)
Huinker and Bill (2017) adapted the eight teaching practices and reimagined them in the
Mathematics Teaching Framework that “allows one to consider the intersectionality among the
eight teaching practices and their relationship to authority, identity, and agency” (p. 245), an
important addition to support deep mathematics learning for all. Figure 1 shows Huinker and
Bill’s (2017) mathematics teaching framework.
While utilizing these practices supports more equitable opportunities for children to
engage in deep mathematics learning (Boaler, 2015), Yolcu (2019) reminded us that despite the
common belief that mathematics as a subject and mathematics learning as a process is neutral, it
is not. Allen (2011) argued that a constructivist approach does not allow for ethnic connections
and lacks ideas for teaching mathematics in a culturally relevant way while adopting social
justice ideas. Instead, Allen (2011) suggested “mathematics as thinking” in which children
collaboratively solve problems that are relevant and meaningful to their lives. Ladson-Billings
(1997) encouraged us to consider that children’s cultural identities and experiences need to be a
part of mathematics learning through culturally relevant teaching. Culturally relevant teaching
has been defined by Ladson-Billings (2009) as “a pedagogy that empowers children
intellectually, socially, emotionally, and politically by using cultural referents to impart
knowledge, skills, and attitudes” (p. 20). This definition highlights the importance of reaching
children through a variety of perspectives and connecting them to their cultural identity.
Similarly, Gay (2010) defined culturally responsive teaching as “using the cultural knowledge,
prior experiences, frames of reference, and performance styles of ethnically diverse children to

27
make learning encounters more relevant to and effective for them” (p. 165). Gay (2002) found
that when teachers understand the cultures of diverse children and build upon their funds of
knowledge, they can support the development of positive identities of children by adapting
curriculum and learning experiences to reflect the cultures of the children.
The equitable mathematics teaching practices framework considered the notion that
mathematics is not a neutral subject and contributed to the conversation about supporting all
children to engage in deep mathematics learning with authority, agency, and a positive identity
with nine practices that support equitable mathematics. These include the following practices:
• Draw on children’s funds of knowledge
• Establish classroom norms for participation
• Position children as capable
• Monitor how children position each other
• Attend explicitly to race and culture
• Recognize multiple forms of discount and language as a resource
• Press for academic success
• Attend to children’s mathematical thinking
• Support development of sociopolitical disposition (Bartell et al., 2017, pp. 11–12)
NCTM (2014)’s mathematics teaching practices and Bartell et al. (2017)’s equitable
mathematics teaching practices as well as other sources of teaching practices that support equity
in mathematics in the literature were considered for what determines equitable EMTPs in this
study. A deep examination of the literature identified several commonalities and places of
intersection that support equitable opportunities for all children to engage in deep mathematics

28
learning while building a positive mathematics identity. Figure 2 displays these places of
intersection.

Figure 2
Equitable Mathematics Teaching Practices

Note. This figure demonstrated the reconceptualization of Huinker and Bill’s (2017) mathematics
teaching framework and Bartell et al.’s (2017) equitable mathematics teaching practices. This
new framework adds power to the current frameworks by aligning those indicators from each
framework that ensure equitable opportunities for deep mathematics learning for all with a focus

29
on developing positive mathematics identities by building on the strengths and identities of all
children. Blue indicates teaching practices from Huinker and Bill (2017) while green indicates
practices from Bartell et al., (2017). The grey indicators are not focused on in this adapted
framework.

As student identity plays a significant role in being able to engage successfully in deep
mathematics learning, in this study, I chose to focus on two dimensions of Huinker and Bill
(2017)’s mathematics teaching framework combined with five of Bartell et al.’s (2017) equitable
mathematics teaching practices that impact identity and development of deep mathematics
learning. They were:
• Implementing tasks that promote reasoning and problem-solving (Huinker & Bill,
2017)
• Facilitating meaningful mathematical discourse (Huinker & Bill, 2017)
• Drawing on students’ funds of knowledge (Bartell et al., 2017)
• Establishing norms for participation (Bartell et al., 2017)
• Positioning students as capable (Bartell et al., 2017)
• Attending explicitly to race and culture (Bartell et al., 2017)
• Attending to students’ mathematical thinking (Bartell et al., 2017)
Adding “other identifying characteristics” to Bartell et al.’s (2017) “attending explicitly
to race and culture” will ensure that other marginalized children such as English Language
Learners or children with disabilities are paid attention to as well. These combined ideas are
referred to as equitable mathematics teaching practices (EMTPs) that promote equitable
opportunities for all children to engage in deep mathematics learning. The next section will

30
describe the EMPTs as well as how they promote equitable opportunities for deep mathematics
learning for all.
Tasks that promote reasoning and problem-solving provide equitable opportunities for all
children to engage in deep mathematics learning and have several important features. To begin,
these tasks should focus on big ideas, principles, and concepts of mathematics as well as the
application of knowledge (English & Gainsburg, 2015). The tasks must be authentic and
meaningful (California Department of Education et al., 2021), allowing all children to make
sense of their world and see how mathematics can be a tool for improving our world (NCTM,
2020). Additionally, to be meaningful, these tasks must consider race, build on and leverage
children’s culture as well as funds of knowledge (Bartell et al., 2017; Suh et al., 2021; Yolcu,
2019). To allow all children to engage in deep mathematics learning, the tasks must also support
a high level of cognitive demand (English & Gainsburg, 2015), ensuring opportunities for sense-
making (Suh et al., 2021), problem-solving, reasoning (Huinker & Bartell, 2017), and
justification through opportunities for collaborative inquiry and discourse (Burdman, 2018). For
all children to be able to access these types of tasks, the tasks must be open to allow the use of
multiple tools, strategies, representations, and solution paths (Bartell et al., 2017; Boaler, 2015;
English & Gainsburg, 2015). Expecting children to utilize one single approach perpetuates
inequities as all children are not in the same place in their learning nor do they think in the same
way. By leveraging children’s strengths and experiences and allowing for all children to utilize
ideas and strategies that make sense to them, these tasks position children as capable and
knowledgeable doers of mathematics (Louie, 2017; Suh et al. 2021). Further, children likely
develop positive mathematics identities as a result (NCTM, 2020).

31
Through such experiences, norms of participation are established (Bartell et al., 2017).
Children come to see that their contributions are valued, and that authority does not rest within
the teacher, but within themselves as meaning makers as together with the entire classroom
community, they decide what counts as appropriate solutions and explanations (Cobb & Hodge,
2002; Gutierrez et al., 1995). In traditional classrooms, the authority often is located within the
teacher or the textbook, where teachers and textbooks define what counts as knowledge and
participation, positioning children as needing to rely on outside sources to validate their ideas.
It is important to note that Lubienski (2000) found evidence contrary to these practices
for supporting equity through reform approaches. She suggested that reform approaches did not
provide fair opportunities for all children. In a study conducted in her classroom, she found that
socioeconomic status may impact children’s preferred methods of teaching. Those of lower
socioeconomic status preferred clearer directions and external support as opposed to those of
higher socioeconomic status who confidently solved problems. As teachers support children, it is
important to know and understand each individual.
Mathematical discourse has been defined by NCTM (2014) as “the purposeful exchange
of ideas through classroom discussion, as well as other forms of verbal, visual, and written
communication” (p. 29). Mathematical discourse can provide opportunities for children to be
powerful contributors to mathematical conversations, positioning all participants as valued and
supporting the development of individual and shared, deep mathematical understanding (Huinker
& Bill, 2017).
Huinker and Bill (2017) conceptualize mathematical discourse as including posing
purposeful questions, using and connecting mathematical representations, eliciting and using
evidence of student thinking, and supporting productive struggle. There are different

32
opportunities for children to engage in mathematical discourse and these serve different
purposes. While children are working individually or collaboratively, the teacher can support
mathematical discourse through questioning, as well as eliciting and using student thinking, to
help move the conversation forward (Munson, 2018). In choosing who’s thinking to consider, the
teacher can challenge spaces of marginality by honoring all contributions as beneficial. In doing
so, teachers are paying attention to race, culture, and other identifying characteristics and
ensuring that any implicit or explicit biases are not impacting whose knowledge they are
advancing.
Teachers can use a variety of assessing and advancing questions to uncover student
thinking and support children in expressing, clarifying, and pushing their thinking (Huinker &
Bill, 2017; Munson, 2018). Teachers must ensure that they do not take over the children’s
thinking with their questions and instead provide scaffolds that allow students to productively
struggle, where the children are challenged, but still have access to the mathematics (Huinker &
Bill, 2017).
Teachers can also support the engagement of children in whole-class discourse which
usually takes place at the end of the lesson. The goal of the whole-class discourse is to build a
shared understanding of the mathematical goals of the lesson (NCTM, 2014). Here, teachers
strategically choose student ideas for examination and organize the conversation so that
mathematical ideas and connections can be highlighted. Smith and Stein (2011) proposed a 5-
practices model for orchestrating deep, mathematically productive conversations that includes
anticipating, monitoring, selecting, sequencing, and connecting. The five practices will be
examined further later in the section titled “Enactment of Equitable Mathematical Teaching
Practices.”

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Mathematical Content Knowledge and Progressions
To provide deep mathematics learning for all children, a robust understanding of content
knowledge (Suh et al., 2021) and knowledge of how the content is situated in learning
progressions (NCSM, 2019) is necessary. This is especially true when teaching through a
constructivist and inquiry approach (Sharp et al., 2011). Thomson and Gregory (2013) found that
teachers with content knowledge gaps had difficulty staying true to reform approaches.
To begin, teachers must understand the mathematics content deeply so that they can plan
cognitively demanding tasks related to the learning goals (Kleickmann et al., 2013). These
learning goals and tasks must be sequenced to form a coherent storyline (Huinker & Bill, 2017).
To do this, teachers must know both where the children have come from mathematically and
where they are going (NCSM, 2019). As math concepts are interrelated and build on each other,
having a deep knowledge of the coherence of content supports deep learning of even more
complex mathematics.
Additionally, Moscardini (2014) found that content and progression knowledge is also a
precursor for engaging with and responding effectively to, children. The author suggested that to
know what you are looking and listening for you must understand the content and progressions
in which they are situated. As EMTPs allow for children to make sense through selecting
strategies and methods that make sense to them, knowing what one is looking and listening for
allows teachers to understand and acknowledge alternative methods proposed by children,
positioning children as capable and competent knowers and doers of mathematics.
Furthermore, to respond effectively and meet the needs of all children, teachers must
know content and progression knowledge deeply (Moscardini, 2014; NCSM, 2019). Teachers
must know where each student is on the progression to provide both scaffolds to allow children

34
to access the deep mathematics as well as push children further if needed. Teachers must also
leverage their deep understanding of the content to provide precise and understandable
explanations (Hill et al., 2019; Schlesinger et al., 2018). Additionally, teachers must utilize their
content knowledge to model concepts visually for children (Hill et al., 2019).
Enactment of Equitable Mathematics Teaching Practice
NCTM (2014) in Principles to Actions asserted, “The question is not whether all students
can succeed in mathematics but whether the adults organizing mathematics learning
opportunities can alter traditional beliefs and practices to promote success for all” (p. 61).
One way the EMTPs can be enacted is through using a three-part lesson, which engages children
in collaborative inquiry (Van de Walle, 2012). The three-part lesson format is a lesson structure
that supports the development of deep mathematics learning for all children through engaging in
authentic, rich tasks that promote reasoning and problem-solving (Huinker & Bill, 2017). Rich
tasks are tasks that have a “high-ceiling and low-floor” meaning that any child can engage in the
task no matter their background knowledge and the task can extend to high levels for those who
need a challenge (Boaler, 2015, p. 2). This provides access to deep mathematics learning for all
children as well as positions them as knowledgeable, moving away from the idea that teachers
must create separate interventions and extensions for children to meet their initial needs. When
tasks are rich, all children are appropriately challenged through the way they approach the tasks
and through the scaffolds and “nudges” provided by the teacher (Munson, 2018).
The 3-part lesson framework consists of a launch, explore, and summary (Van de Walle,
2012). The launch portion of the 3-part framework is where children are introduced to a task, the
learning goal is understood, prior knowledge is activated, and clear expectations for work are
established. Children discuss ideas and make a plan for how they might approach the problem.

35
The explore portion is a time for teachers to release responsibility to the children and
allow them to engage in productive struggle (Huinker & Bill, 2017). During this time, teachers
support discourse between small groups of children who are working collaboratively to solve the
problem with strategies, tools, and solution paths of choice (Huinker & Bill, 2017). This allows
all children to approach problems in ways that make sense to them and positions them as
capable. Teachers attend to children’s mathematical thinking (Bartell et al., 2017) and pose
purposeful questions (Huinker & Bill, 2017) that maintain the cognitive demand of the task,
support productive struggle (Huinker & Bill, 2017), and again, position children as capable and
knowledgeable doers of mathematics. Also with these questions, teachers either support or
extend children’s thinking so the children may access the task or go beyond. Additionally,
teachers elicit and use student ideas to move the lesson forward, as well as choose children’s
work to examine in the summary. To do this, teachers must attend to their possible biases based
on race, culture, or other factors (Bartell et al., 2017) to ensure that they are not valuing the
knowledge of some over others.
The summary is the last part of the 3-part lesson. The goal of the summary is to build a
shared understanding of the mathematics intended for the learning goals (Huinker & Bill, 2017).
This is done through a whole group discussion of ideas elicited from the children (Smith & Stein,
2018). To orchestrate a productive discussion, the teacher must have anticipated and monitored,
then strategically selected and sequenced student work to be discussed. Ideas chosen might be
various strategies or representations, or even misconceptions that occur with many children
during the exploration. For all children to have access, the teacher might choose a strategy that is
easily understood or a representation that is more concrete to start with, then build towards the
goals of the lesson. For example, a teacher may select children who used the same strategy but

36
used different modes of representation such as cubes, drawings of objects, number lines, and tape
diagrams. The teacher would start with the cubes, then build from there, supporting student
discourse around the similarities and differences between the representations to build deeper
mathematical connections and understanding (Huinker & Bill, 2017). Sometimes misconceptions
are shared to support children in analyzing their errors or misunderstandings and to learn from
mistakes. In turn, children develop a growth mindset (Dweck, 2006). The teacher’s role is not to
confirm right or wrong, but to facilitate discourse between children and ask purposeful questions
that support further engagement in deep mathematics learning (Huinker & Bill, 2017). By
examining children’s ideas and those of others, children’s ideas are validated (Berry, 2019). This
helps children to see themselves as powerful knowers and doers of mathematics (NCTM, 2020).
Table 2 illustrates elementary teachers’ knowledge influences on the enactment of EMPTs and
the related literature.

37
Table 2
Summary of Assumed Knowledge Influences on a Grade 3–5 Teacher’s Ability to Enact EMTPs
Assumed knowledge
influence
Research literature
Factual and conceptual
knowledge
Teachers know and
understand goals of
mathematics and
mathematics teaching.
Boaler, 2008; California Department of Education et al., 2021;
English & Gainsburg, 2015; Gutstein, 2012, NCTM, 2020;
Yolcu, 2019
Teachers know and
understand what
equitable math teaching
practices are.
Aguirre et al., 2013; Allen, 2011; Anthony et al., 2015; Bartell
et al., 2017; Boaler, 2015; Burdman, 2018; California
Department of Education et al., 2021; Cobb & Hodge, 2002;
English & Gainsburg, 2015; Gay, 2010; Gutierrez & Irving,
2012; Gutierrez et al., 1995; Huinker & Bill, 2017; Ladson-
Billings, 1997; Ladson-Billings, 2009; Louie, 2017;
Lubienski, 2000; Munson, 2018; NAEYC, 2019; NCTM,
2014; NCTM, 2020; National Governors Association Center
for Best Practices, Council of Chief State School Officers,
2010; Smith & Stein, 2011; Suh et al., 2021; Szabo et al.,
2020; Tan et al., 2019; Wachira & Mburu, 2017; Windschitl
et al., 2011; Yolcu, 2019
Teachers know and
understand
mathematical content
knowledge and
progressions.
Thomson & Gregory, 2013; Hill et al., 2019; Huinker & Bill,
2017; Kleickmann et al., 2013; Moscardini, 2014; NCSM,
2019; Schlesinger et al., 2018; Sharp et al., 2011; Suh et al.,
2021
Procedural knowledge
Teachers know how to
enact equitable math
teaching practices in
their classrooms.
Bartell et al., 2017; Berry, 2019; Boaler, 2015; Dweck, 2006;
Huinker & Bill, 2017; Munson, 2018; NCTM, 2014; NCTM,
2020; Smith & Stein, 2018; Van de Walle, 2012

Motivation
To accomplish something, one must be motivated (Rueda, 2011). Schunk et al., (2009)
have suggested that motivation is impacted by both internal factors and external factors.

38
Additionally, the authors suggested three indicators of motivation. These include active choice,
persistence, and effort. When supporting people to attain goals, factors that impact motivation as
well as the three indicators must be considered (Rueda, 2011). There are several dimensions of
motivation. For this study, efficacy was examined.
In addition to knowledge, teachers must be motivated to want to teach in ways that
support all children to have opportunities to engage in deep learning of mathematics. Teacher
efficacy and collective efficacy are two theories or constructs that apply to elementary
mathematics teachers being able to enact EMTPs. These will be discussed in the next sections.
Self-efficacy, the belief that one can perform a task, maintains a strong impact on
motivation (Schunk, 2020). There are several indicated influences on self-efficacy. To begin,
observing different peer models likely increases positive outcome expectations resulting in an
increased sense of self-efficacy (Schunk, 2020). Additionally, small group interactions can
increase self-efficacy because as knowledge is shared and group members have a responsibility,
group members tend to feel successful and social comparisons are reduced. In this sense, the
situation is non-threatening because group members can build on the collective knowledge of the
group rather than compare their abilities to others.
Building on the idea of self-efficacy is teacher efficacy, a construct that refers to a
teacher’s belief that they can help students learn (Schunk & DiBenedetto, 2016). Teacher
efficacy is important for motivation specifically to teaching. Those teachers with low self-
efficacy for teaching tend to avoid tasks they don’t think they can accomplish while those with
high teacher efficacy are willing to take on teaching challenges. This could have a significant
impact on ensuring that all children have opportunities for deep mathematics learning. For
example, if a teacher does not understand reform approaches to mathematics teaching, they may

39
not feel capable of teaching through such approaches, resulting in a lack of willingness to learn
about or try different approaches.
Collective efficacy is a different, yet similar construct that is important for this study.
Collective efficacy refers to the “perceptions of a group of teachers in a school that their efforts
as a whole will positively affect students” (Goddard et al., 2000). Goddard et al., (2000)
suggested that collective efficacy could likely be improved by teachers working collaboratively
to learn, problem-solve, and successfully implement change together. Voelkel and Chrispeels
(2017) found that when teachers envisioned themselves as a high-functioning professional
learning community (PLC), increases in the perceived sense of collective efficacy were likely to
be experienced. For teachers to feel efficacious, they must collaboratively engage in mastery
experiences, or those experiences that make them feel successful (Loughland & Nguyen, 2020).
They noted both direct and vicarious learning experiences contribute positively to collective
efficacy but that the sources of collective efficacy are important. Some sources that were found
to lead to mastery experiences, and thereby increase collective efficacy, were detailed,
collaborative lesson planning, reflective conversations, and mentoring in conjunction with an
expert. Table 3 shows elementary teachers’ motivation influences related to the enactment of
EMPTs and the relevant literature.

40
Table 3
Summary of Assumed Motivation Influences on a Grade 3–5 Teacher’s Ability to Enact EMTPs
Assumed motivation influence Research literature
Teacher efficacy
Teachers need to believe that they are capable of
enacting equitable teaching practices in their
classrooms.
Rueda, 2011; Schunk et al., 2009;
Schunk & DiBenedetto, 2016;
Schunk, 2020
Collective efficacy
Teachers need to believe that their collective effort
will positively affect student learning in
mathematics.
Goddard et al., 2000; Loughland &
Nguyen, 2020; Voelkel & Chrispeels,
2017

Organization
An organization’s culture is made up of both the cultural settings and models (Gallimore
& Goldenberg, 2001). Cultural models refer to the “shared mental schema or normative
understanding of how the world works, or ought to work” (Rueda, 2011, p. 55). Alternatively,
cultural settings are the everyday ways in which organizational policies are enacted. Examining
the cultural settings and models of an organization can give us insight into the way the
organization operates and why organizational goals may not have been met (Rueda, 2011).
The literature revealed several potential sources of organizational factors that support the
organizational goal to enact EMTPs. For this research, I focused on having a common vision and
philosophy, ongoing professional learning opportunities, and standards-based curriculum
instructional materials with implementation support.
Common Vision and Philosophy

41
A common vision is essential to direct change (Kotter, 2011). Additionally, having a
common vision supports team-goal commitment, resulting in employees' improved cooperation
which, in turn, helps them develop collective efficacy (Chai et al., 2017).
To create an effective and equitable mathematics program, every teacher must be clear on
what effective and equitable teaching and learning of mathematics looks and sounds like
(NCSM, 2019). Additionally, teachers must understand what is expected for all children in terms
of experience. A common vision, which is shared, and which teachers develop a common
understanding of, can promote an effective mathematics program. The vision should provide a
clear direction for teachers and leaders and inspire action toward common goals, provide focus,
as well as provide support with strategic planning and professional development.
Ongoing Professional Learning
There has been a great deal of research on features of professional learning with mixed
perspectives regarding what is effective. This might be attributed to the different measurement
tools generally being utilized by large- and small-scale studies (Copur-Gencturk & Thacker,
2021). Most large-scale studies utilize self-reports while small-scale studies tend to rely more on
direct assessments and observation. Both types of studies have revealed effective features of
professional development. However, Copur-Gencturk and Thacker (2021) found discrepancies
between self-reports and direct assessments. As such, some features that have been found
effective as indicated by self-report, may not be entirely accurate. Nonetheless, there have been
promising findings that warrant consideration.
The first large-scale empirical comparison of effective professional development
characteristics focused on the core features of professional learning and the structures of
professional learning (Garet et al. 2001). Findings based on teacher self-reports revealed that

42
sustained and intensive professional learning, a focus on academic subject matter, opportunities
for active learning, and coherence of professional learning experiences led to increased teacher
knowledge.
In “Professional Learning in the Learning Profession: A Status Report of Teachers’
Development in the United States and Abroad” authors acknowledged some similarly reported
positive effects when synthesizing global research (Darling-Hammond et al., 2009). Intensive
and ongoing professional learning connected to practices had a greater chance of influencing
teaching practices than less intensive, one-off professional development sessions that focused on
theory. This included the application of knowledge in both planning and instruction.
Additionally, a focus on teaching and learning of academic content which allowed everyday
challenges faced by teachers to be addressed was found to be impactful. This includes things
such as modeling lessons and analyzing children’s performance data to identify common errors
and misconceptions in student thinking, determining proficiency among teachers, and
determining effective and ineffective instructional strategies.
Other positive professional learning themes were found as well in this report. Work done
in professional learning sessions being connected to other school initiatives was important as it
allowed teachers to easily implement new strategies since they could see how they were
connected and not an isolated learning event or skill acquired. Additionally, collaborative
approaches were an important theme noted. The report suggested that collaborative work
allowed for change beyond the individual classrooms and strong working relationships, allowing
for greater consistency in practices and a willingness to share and try new practices.
Drilling down into more specifics, other researchers have explored a variety of structural
considerations and environment indicators, collaborative approaches to professional

43
development, and the content of the professional development sessions and their impact on
teacher learning. In other words, the ‘why,’ the ‘how,’ and the ‘what.’ Many findings are like the
large-scale findings shared previously, others offer different or contain more specific insights.
Similar to findings mentioned previously, other researchers found that ongoing, long-
term, coherent, active learning engagements (Darling-Hammond et al, 2009; Felton-Koestler,
2019; Rosenquist et al., 2015) where teachers bring together theory and practice (Chong &
Kong, 2012; Kennedy, 2016) were necessary for teacher learning. Also noted was the importance
of a low stakes, safe environment where teachers could build trust and relationships with peers
and facilitators through engaging in discussions, having the ability to disagree (Felton-Koestler,
2019), challenging beliefs about the nature of mathematics and who can learn mathematics, as
well as addressing positionality (Louie, 2020; Yolcu, 2019). Professional development must also
be coordinated between leaders, coaches, and/or specialists, and teachers to ensure consistency
(Jackson & Cobb, 2013).
Research also indicated the importance of teacher collaboration in learning (Gasser,
2011; Harris et al., 2017; Kleickmann et al., 2013; Lin et al., 2013; Loughland & Nguyen, 2020;
Louie, 2020). Underpinning this notion is social constructivism, the idea that learning is socially
situated, and that knowledge is constructed through social negotiation (Vygotsky, 1962).
Rosenquist et al., (2015) found that collaboration was most impactful when done in stable groups
of teachers. Louie (2020) stated that collaborative work groups should be grounded in
discussions that promote student agency, an important construct for equity. While teachers do
not often exclude children intentionally, it was found that “much of their instruction had the
unintended effect of reinscribing the culture of exclusion” (p. 513). Explicit, ongoing reframing
of thinking about who is included and excluded in classrooms through collaboratively examining

44
positionality and beliefs is important for learning to provide equitable opportunities for all
children.
Many collaborative structures have been found to support teacher learning based on self-
reports. Collaborative reflection on practice, both during and after teaching episodes in the form
of video analysis (Celedon-Pattichis et al., 2018), content-focused coaching (Gibbons & Cobb,
2016), sequenced cycles of investigations and action (Rosenquist et al., 2015), and mentoring
(Loughland & Nguyen, 2020) have been cited as potentially useful for improving practice.
Informal learning interactions such as conversations between peers have also been found to be
supportive in teachers’ self-reported learning (Kleickmann et al., 2013). Including noticing of
children’s knowledge, participation, and power in collaborative conversations can increase
access to opportunities for deep math learning for all by supporting the development of an asset
mindset of teaching (Celedon-Pattichis et al., 2018).
While the ‘how’ and “why’ have been discussed, a final consideration for teacher
collaborative learning is the “what.” Rosenquist et al. (2015) suggested that there must be a clear
focus on content and student learning. This might include learning about, discussing, and
reflecting on teaching knowledge, for example, specific instructional practices or routines such
as single high-leverage tasks and discourse practices (Celedon-Pattichis et al., 2018).
Additionally, discussions specific to mathematical content knowledge, curricular knowledge
(Copur-Gencturk, 2012), and knowledge of how children learn mathematics (Copur-Gencturk,
2019) are important. Such discussions might include an understanding of math content standards,
their progressions over time, and connections across other mathematical and non-mathematical
domains commonly referred to as ‘coherence’ (NCTM, 2020), discussions about common
misconceptions, student solutions, and strategies (Copur-Gencturk, 2019) as well as how to

45
respond to these, or how to utilize curriculum instructional materials with integrity (NCTM,
2020).
Standards-Based Curriculum Instructional Materials With Support for Implementation
For teachers to enact EMTPs, the organization must provide supportive physical
resources that align with the vision and philosophy (Hill et al., 2008; NCTM, 2020).
Additionally, as mathematics is cumulative and children deepen their understanding over time,
attention to the coherence of the standards-aligned curriculum is necessary. As such, curricular
instructional materials that ensure instructional experiences build and connect provide an
important framework for teaching (NCTM, 2020).
There is evidence that curriculum instructional materials can have a positive impact on
teaching when the materials provide appropriate supports including but not limited to providing
clarity of learning goals and how these are supported by the learning engagements, helping to
attend to children’s thinking, and helping teachers understand design choices through making
them visible. However, teachers must read them (Charalambous & Hill, 2012; Remillard et al.,
2019). Quality resources also support content knowledge development. Schneider and Krajcik
(2002) referred to these materials as “educative.” Remillard et al. (2019) argued that these
materials should be a source of professional learning for teachers. The intent of these “educative”
supports is the flexible and productive use of resources by teachers.
However, just providing an educative, standards-based resource will not improve the
quality of teaching, especially due to the increased demands on teachers when implementing
EMTPs (Charalambous & Hill, 2012). Implementing a curriculum is more than following the
curricular instructional materials. While curricular instructional materials can guide teachers,
teachers must utilize curriculum instructional materials with integrity rather than fidelity,

46
meaning that a teacher should use professional expertise to stay true to the big ideas of the
curriculum and philosophy, while making adjustments to the instructional materials to
accommodate children’s strengths, needs, interests, culture, and language (Huinker & Bill,
2017). Ball and Cohen (1996) referred to this work that teachers do while teaching as
“curriculum enactment” (p. 7). Remillard et al. (2019) suggested that this work includes listening
to, interpreting, and responding to children’s responses to move them towards determined
learning goals. This also requires teachers to see opportunities and limitations contained within
the resources. Jacobs et al. (2010) refer to this as “curricular noticing.” Curricular noticing is
done in phases: curricular attending, curricular interpreting, and curricular responding. These
skills enable teachers to recognize, make sense of, and strategically utilize the opportunities
provided within the materials. Curricular noticing is impacted by a teacher's educational goals
and what the teacher sees as the purposes of interactions within the classroom (Sherin & Drake,
2009). If a teacher views their role as directly teaching strategies to children, they will pay
attention to opportunities in the materials to provide those types of experiences.
To effectively notice and enact curricular resources with integrity, teachers need to have
productive beliefs that support EMTPs, knowledge of mathematics content and progressions,
knowledge of children’s thinking and how to respond and prompt to support the movement
towards learning goals (Remillard et. al, 2019), pedagogical approaches and teaching strategies
that support the pedagogical approaches, as well as an understanding of the curricular materials
to make informed decisions (Huinker & Bill, 2017). Charalambous and Hill (2012) found that for
those teachers with low mathematical knowledge for teaching (MKT), which includes
specialized content knowledge and teaching strategies, unsupportive curricular materials could
negatively impact instruction. Alternatively, those with high MKT were able to mitigate the

47
unsupportive nature of the materials and were able to make adjustments to the resources that
better supported student learning.
Charalambous and Hill (2012) also reported that teachers need focused and supported
opportunities to learn how to utilize materials with integrity. Lampert et al. (2011) suggested this
is best done through “social practice,” what teachers “do routinely with their colleagues as part
of a structured social system working on a joint enterprise using a common set of resources to
meet common objectives” p. 1367. They found that “social practice” allowed teachers to
examine their teaching practices and engage in instructional problem-solving rather than blaming
children for their performance. The use of the common resource was of utmost importance
because teachers had a common referent with which to discuss teaching practices and problems
of practice using a shared language and artifacts. Table 4 outlines the organization's influence on
an elementary teacher’s ability to enact EMPTs and the related literature.

48
Table 4
Summary of Assumed Organizational Influences on a Grade 3–5 Teacher’s Ability to Enact
EMTPs
Assumed organizational influence Research literature
The organization needs to provide a
common vision and philosophy to
support an elementary teacher’s
enactment of EMTPs.
Chai et al., 2017; Kotter, 2011; Loughland & Nguyen,
2020; NCSM, 2019
The organization needs to provide
ongoing professional learning to
support an elementary teacher’s
enactment of EMTPs.
Celedon-Pattichis et al., 2018; Chong & Kong, 2012;
Copur-Gencturk, 2012; Copur-Gencturk & Thacker,
2021; Darling-Hammond et al., 2009; Felton-
Koestler, 2019; Garet et al. 2001; Gasser, 2011;
Gibbons & Cobb, 2016; Harris & Jones, 2017;
Jackson & Cobb, 2013; Kennedy, 2016; Kleickmann
et al., 2013; Lin et al., 2013; Loughland & Nguyen,
2020; Louie, 2020; NCTM, 2020; Rosenquist et al.,
2015; Vygotsky, 1962; Yolcu, 2019

The organization needs to provide
standards-based curriculum
instructional materials with
support for implementation to
support an elementary teacher’s
enactment of EMTPs.

Ball & Cohen, 1996; Charalambous & Hill, 2012;
Sherin and Drake, 2009; Hill et al., 2008; Huinker &
Bill, 2017; Jacobs et al. 2010; Lampert et al., 2011;
NCTM, 2020; Remillard et al., 2019; Schneider &
Krajcik, 2002

The above-mentioned knowledge, motivation, and organization factors in this Chapter 2
will be used as a foundation for data collection in Chapter 3. The next section will detail the
interaction between the knowledge, motivation, and organizational influences impacting the
enactment of EMTPs.

49
Conceptual Framework: The Interaction of Stakeholders’ Knowledge and Motivation and
the Organizational Context
There are several knowledge, motivation, and organizational influences that impact
equitable opportunities for all children to engage in deep mathematics learning for all. Figure 3
displays how these influences interact to support this goal. Following Figure 3 is a detailed
description of these interactions.

Figure 3
Conceptual Framework: Equitable Opportunities for All Children to Engage in Deep
Mathematics Learning

50
Many studies have found the presence of inequitable opportunities for the development of
deep mathematical learning for all children, resulting in a lack of conceptual understanding and
oftentimes, a negative mathematical identity for some (NCTM, 2020). As such, in this concept
map, equitable opportunities for all children to engage in deep mathematics learning are
centered, with the hopes of transforming all children into capable knowers and doers of
mathematics with positive mathematical identities. This outcome is highlighted by a
unidirectional arrow from teacher knowledge and motivation to the outcome.
Situated directly below the outcome is the teacher knowledge and motivation necessary
to enact EMTPs because teachers have a direct impact on student learning. Teacher knowledge
and motivation are connected through ongoing arrows indicating the influence between each of
these factors. To enact EMPTs, teachers need to first understand the goals of mathematics
including deep mathematical understanding, joy and wonder of mathematics, creating capable
knowers and doers of mathematics, and understanding and critiquing the world (NCTM, 2020).
Further, teachers must have a strong understanding of mathematics content and progressions to
be able to support children where they are and address misconceptions (Kleickmann et al., 2013).
As teachers collectively develop shared knowledge of content and EMTPs and work together to
learn, problem-solve, and successfully implement change, they are likely to become more self-
efficacious (Goddard et al., 2000). When teachers’ efficacy is high, they feel more positive about
their practice and feel more motivated to work hard at a task, in this case learning about or
enacting EMTPs (Rueda, 2011). In turn, this increases collective efficacy as well (Loughland &
Nguyen, 2020).
On the outside of the concept map, enveloping the outcomes and teachers, is the
organization, suggesting that the organization plays a key role in supporting teachers to develop

51
the motivation and knowledge needed to meet goals (Rueda, 2011). Many organizational
influences support teacher knowledge and motivation for the enactment of EMTPs; however, I
have chosen to focus on a common vision and philosophy, standards-based curriculum
instructional materials with support, and ongoing professional learning as these are three key
factors in enactment. The following sections will describe these influences. While culture and
leadership are key factors in supporting educational goals, they are beyond the scope of the
study.
A common vision is essential to direct change (Kotter, 2011). A unidirectional arrow can
be found from the common vision and philosophy towards aligned resources as resources with
this vision and philosophical underpinnings need to be selected (Hill et al., 2008). A
unidirectional arrow can also be found coming from this organizational factor toward teacher
knowledge and motivation, speaking to the idea that a common vision and philosophy will
determine the teacher knowledge required to act on the vision. Additionally, having a common
vision supports team-goal commitment, resulting in employees' improved cooperation which, in
turn, helps them develop collective efficacy (Chai et al., 2017).
Ongoing professional learning opportunities and feedback must occur for teachers to be
successful. This professional learning should support teachers with theory and be directly related
to the practice in the classroom (Chong & Kong, 2012). Professional development must be
coordinated between leaders, coaches and/or specialists, and teachers to ensure consistency
(Jackson & Cobb, 2013). The bidirectional arrow between professional development and teacher
knowledge and motivation demonstrates that not only does professional learning influence
teacher knowledge and motivation, but professional learning experiences must also be responsive

52
to the current levels of knowledge and motivation to scaffold appropriate teacher learning
(Ertmer & Simons, 2005).
For teachers to enact EMTPs, the organization must also provide standards-based
curriculum instructional resources that align with the vision and philosophy (Hill et al., 2008;
NCTM, 2020). The bidirectional arrow indicates the influence of the resources on professional
learning, including the resource as a source of professional learning (Dietiker et al., 2018;
Remillard et al., 2019), and the influences of the professional learning on the resource enactment.
The bidirectional arrow from resources to teacher knowledge and motivation indicates a mutual
influence as well. Quality resources support teacher knowledge of EMTPs and how to enact
them as well as support content knowledge development. Teacher knowledge influences resource
enactment as well. Charalambous and Hill (2012) found that those with a high level of content
knowledge as well as knowledge about how to teach mathematics were able to adjust the
resources to better support student learning. Having a resource that supports the vision and
philosophy influences teacher efficacy as teachers can develop shared knowledge (Goddard et
al., 2000). Additionally, as Charalambous and Hill (2012) stated, teachers must have focused and
supported opportunities to learn how to utilize the materials.
Summary
Opportunities for deep mathematical learning for all children remain inequitable. Despite
repeated attempts to reform approaches to mathematics teaching in the United States, some
children, especially those from marginalized populations, remain engaged in a narrow
curriculum with traditional teaching practices that support the idea that the purpose of
mathematics teaching is to transmit a fixed body of knowledge and that deep mathematics

53
learning is only for some. This often results in low academic achievement and children who do
not view themselves as knowers and doers of mathematics.
To better understand the factors contributing to persistent inequitable opportunities for
deep mathematics learning, Chapter 2 outlined the literature that was examined to understand the
historical perspectives of inequities in mathematics, what equity means in mathematics, and the
causes of inequities. This was followed by the knowledge, motivation, and organizational
influences that impact the teacher’s ability to enact practices that support equitable opportunities
for all children to engage in deep learning of mathematics and likely develop a positive
mathematical identity.
Finally, Chapter 2 concluded with a conceptual framework showing the relationships
between the knowledge, motivation, and organizational influences that impact the enactment of
EMTPs. Chapter 3 will present the study’s methodological approach used, demonstrating how
the study gained an understanding of the current levels of Grades 3–5 teachers’ knowledge and
motivation related to the enactment of EMTPs and how the organization is both supporting and
hindering this enactment.

54
Chapter Three: Methodology
The purpose of this study was to conduct an adapted gap analysis to examine the
knowledge, motivational, and organizational influences impacting the successful enactment of
equitable mathematics teaching practices (EMTPs) so that children in Grades 3–5 are afforded
opportunities to engage in deep mathematics learning and to develop positive mathematics
identities. While a complete adapted gap analysis would have focused on leaders, children, and
all elementary mathematics teachers, for practical purposes, Grades 3–5 mathematics teachers
were the focus of this analysis. The analysis began by generating a list of possible or assumed
influences determined through a review of related literature that was examined systematically to
focus on actual or validated causes.
As such, the questions that guided this study were as follows:
1. What is the current status of teachers’ knowledge and motivation related to enacting
equitable mathematics teaching practices, or those teaching practices that provide
access to deep mathematics learning for all learners?
2. How do organizational factors influence teachers’ capacity to enact equitable
mathematics teaching practices?
Conceptual and Methodological Framework
Clark and Estes Gap Analysis Framework
The Clark and Estes (2008) framework is a problem-solving process used to study
stakeholder performance within an organization. It provides a way to clarify stakeholder and
organizational goals, assess performance or achievement of the goals, and describe the gaps in
stakeholder performance with regards to organizational goals. This framework helps to identify
assumed performance causes in the areas of knowledge, motivation, and organization.

55
Knowledge assessments, motivation scales, and organization perception scales are used as
measurement tools. The process supports the validation of root causes of identified gaps
followed by defining solutions, implementation, and evaluation of solutions. A flow chart of the
gap analysis process is provided in Figure 4.

Figure 4
Gap Analysis Overview

Note. Adapted from Turning Research into Results: A Guide to Selecting the Right Performance
Solutions (p. 22) by R.E. Clark and F. Estes, 2008, CEP Press. Copyright 2008 by Richard E.
Clark and Fred Estes.

56
In this study, the Clark and Estes framework was adapted as an improvement study.
Measures determined the level of knowledge, motivation, and organization in the improvement
of the current study.
Adaptation of the Framework: Improvement Model
Green View River International School’s (GVRIS) performance problem at the root of
this study was the inconsistent enactment of EMTPs, resulting in inequitable opportunities for all
children in Grades 3–5 to engage in deep mathematics learning. As such, this study utilized an
improvement model to find a strategy to improve the teaching of mathematics for all children
utilizing EMTPs. Based on the findings of teacher knowledge and motivation, and their
perceptions of how the organization supports or hinders the enactment of EMTPs,
recommendations for improvement have been made.
Assessment of Performance Influences
Several knowledge, motivation, and organizational influences related to the enactment of
EMTPs in Grades 3–5 were found upon examination of the literature. The following sections
will describe the relevant influences and how they will be assessed within this study.
Knowledge Assessment
Four possible knowledge influences were revealed through an examination of the
relevant literature. Three factual and conceptual knowledge influences include: Teachers must
know and understand goals for mathematics teaching; Teachers must know and understand what
equitable math teaching practices are; and Teachers must know and understand mathematical
content knowledge and progressions. All three factual and conceptual influences were assessed
through interviews. Goals for mathematics, as well as content knowledge and progressions, were
assessed through observations as well. This study included one procedural knowledge influence:

57
Teachers must know how to enact equitable math teaching practices in their classrooms.
Knowledge of enactment of equitable math teaching practices was assessed through observations
and interviews. Table 5 identifies the method of assessment for each of the knowledge
influences. Influences with a limited number of questions in the interview protocol have been
included in the table. For those influences containing several questions in the interview protocol,
a brief overview of the types of questions has been provided with one or two examples of
questions. A complete list of questions can be found in Appendix A: Interview Protocol for
Teachers. Similarly, as the observation protocol is extensive, a brief overview of the types of
observable teaching behaviors has been provided with one or two examples of observable
teaching behaviors. A complete list of observable teaching behaviors can be found in Appendix
B: Observation Tool.

Table 5
Summary of Knowledge Influences and Method of Assessment
Assumed knowledge
influence
Interview item Observation item
Factual and
conceptual
knowledge
Teachers know
and understand
the goals of
mathematics and
mathematics
teaching.
In your opinion, what should the
goals of mathematics be?
• Why do we teach
mathematics?
• What are you trying to
accomplish in your math
lessons?
In your opinion, what does it
mean to be successful in
mathematics?
Teacher positions students as
knows and doers of
mathematics.
Teachers pose problems in a way
that invites curiosity and joy.
Tasks posed support
understanding, critiquing, and
changing the world.
Teachers know
and understand
what equitable
Topics will cover personal
definition of equity in the
mathematics classroom,
practices that support equitable

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Assumed knowledge
influence
Interview item Observation item
math teaching
practices are.
mathematics teaching, and
student identity. i.e., In your
opinion, what teaching
practices support an equitable
mathematics classroom?
See Appendix A for further
questions.

Teachers know
and understand
mathematical
content
knowledge and
progressions.
Topics will cover big ideas,
concept development,
strategies, representations, and
misconceptions. i.e., Please
describe the big ideas of Place
Value. What do children need
to understand? How do
children develop the concepts
of Place Value over time?
See Appendix A for further
questions.
Teacher anticipates, notes, and
fully addresses common
misconceptions.
When teacher engages with
students through questioning,
clarifying, or choosing student
solutions for presentation,
mathematical content is correct
and precise.

Procedural
knowledge
Teachers know
how to enact
equitable math
teaching
practices in their
classrooms.
Topics will cover description of
what teachers and children
would be doing during a
lesson, teaching style, beliefs
about problem-solving and
constructing arguments, and
how to address misconceptions.
ie. Suppose I was in your
classroom for a typical math
lesson. What would I see you
doing? What would I hear you
saying? What would the
children be doing? What would
the children be saying?
See Appendix A for further
questions.
Observations will focus on tasks,
discourse, norms of
participation, meeting children’s
needs, content, and attending to
race, culture, and other
identifying characteristics. ie.
Nature of the mathematical tasks
is rich, appropriately
challenging, complex, and lends
to multiple entry points and
solution pathways. Tasks are
posed in ways that invite
speculation.

Teacher appears to have
established a protocol/norm for
the learning culture where
children are expected to
participate and are positive and
supportive towards each other.
See Appendix B for further
observable teaching behaviors.

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Motivation Assessment
Two possible motivation influences were revealed through examination of the relevant
literature. Two motivational influences include: Teachers need to believe that they are capable of
enacting equitable teaching practices in their classrooms (teacher efficacy); and Teachers need to
believe that their collective effort will positively affect student learning in mathematics
(collective efficacy). Both motivation influences were assessed through interviews. Table 6
identifies the method of assessment used for each of the motivation influences. Influences with a
limited number of questions in the interview protocol have been included in the table. For those
influences containing several questions in the interview protocol, a brief overview of the types of
questions has been provided with one or two examples of questions. A complete list of questions
can be found in Appendix A: Interview Protocol for Teachers.

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Table 6
Summary of Motivation Influences and Method of Assessment
Assumed motivation influence Interview item
Teacher efficacy
Teachers need to believe
that they are capable of
enacting equitable
teaching practices in their
classrooms.
Topics will cover feelings about ability to teach mathematics
to all children and perceived contributing factors to
children’s success or struggle in mathematics. Ie. How do
you feel about your ability to teach mathematics to all
children? Follow-up – What about those children whose
first language is not English, who have a diagnosed
disability, or who learn math a different way at home?
See Appendix A for further questions.
Collective efficacy
Teachers need to believe
that their collective effort
will positively affect
student learning in
mathematics.
The school expects teachers to work in professional learning
communities (PLCs) to collectively meet the needs of all
children. How confident are you that together with your
team, all children in G_, including those of different racial
groups, those who are ELLs, or those with identified
learning disabilities, will be successful in mathematics?
Please describe how your team supports each other in
collectively figuring out how to best support your children,
if at all.

Organization/Culture/Context Assessment
Three possible organization influences were revealed through examination of the relevant
literature. Three possible organizational influences include: The organization needs to provide a
common vision and philosophy to support an elementary teacher’s enactment of EMTPs; The
organization needs to provide ongoing professional learning to support an elementary teacher’s
enactment of EMTPs; and the organization needs to provide standards-based curriculum
instructional materials with support for implementation to support an elementary teacher’s
enactment of EMTPs. All organizational influences were assessed through interviews. Table 7
identifies the method of assessment for each of the organizational influences. Influences with a

61
limited number of questions in the interview protocol have been included in the table. For those
influences with several questions in the interview protocol, a brief overview of the types of
questions has been provided with one or two examples of questions. A complete list of questions
can be found in Appendix A: Interview Protocol for Teachers.

Table 7
Summary of Organization Influences and Method of Assessment
Assumed organizational influence Interview item
The organization needs to provide a
common vision and philosophy to
support an elementary teacher’s
enactment of EMTPs.
Suppose I was a new teacher to the school and I
asked you to describe what SAS believes is
important in teaching and learning of mathematics.
What would you say?
Describe your math experience as a teacher at SAS.
What might be some contributing factors to this?
The organization needs to provide
ongoing professional learning to
support an elementary teacher’s
enactment of EMTPs.
Topics will include how the school and team support
the teacher, describing the ideal organization, and
suggestions for improvement. ie. Describe how
your school supports your role as an elementary
mathematics teacher, if at all? How has this
impacted your practice, if at all?
Please describe how your team supports each other in
collectively figuring out how to best support your
children, if at all.
See Appendix A for further questions.
The organization needs to provide
standards-based curriculum
instructional materials with support
for implementation to support an
elementary teacher’s enactment of
EMTPs.
Describe your math experience as a teacher at SAS.
What might be some contributing factors to this?
Describe how your school supports your role as an
elementary mathematics teacher, if at all? How has
this impacted your practice, if at all?

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Participating Stakeholders and Sample Selection
The stakeholder group of focus for this paper was Grades 3–5 classroom teachers.
Teachers from a grade band rather than K–5 were selected primarily due to time constraints.
Grades 3–5 were selected rather than K–2 because the complexity of the mathematics in Grades
3–5 is greater to K–2, building on the foundations of the primary grades. To gain various
perspectives within the grade band, I sampled Grades 3–5 elementary mathematics teachers
through two sets of criteria. Six teachers were selected in total. This gave me varied information
but allowed for quality data collection due to time constraints.
All teachers selected must:
• be classroom teachers in Grades 3–5
• have taught at the school for a minimum of 3 years to understand how the school
operates
In addition to the above criteria, two teachers were selected from each of the two following
categories based on knowledge from personal conversations and coaching work conducted by the
instructional coaches for Grades 3–5:
• teachers who utilize traditional methods of teaching mathematics (e.g., demonstrating
how to solve problems then having the children duplicate what the teacher does, only
allowing for one way to solve problems, relying on rote practices rather than
understanding, etc.)
• teachers who utilize EMTPs
Recruitment
To gain access to and recruit teachers, I first met with the principal to explain the scope
of the study including sharing the selection criteria and relevant documents such as informed

63
consent (Lochmiller & Lester, 2017). Once permission was granted, I then emailed the
representative teachers with a letter explaining the scope of the project to gain interest (See
Appendix E: Recruitment Letter). While Lochmiller and Lester (2017) suggested that the
principal emails documents to the potential participants, I corresponded with teachers directly
because of the nature of relationships at the school.
Instrumentation
Data collection took place through two observations of classroom practice and semi-
structured interviews which allowed for flexibility (Lochmiller & Lester, 2017). A purposeful
sample of Grades 3–5 elementary mathematics teachers was engaged. An observation protocol
was utilized. While it is known that teacher self-reports often do not match direct assessments
(Copur-Gencturk & Thacker, 2021), comparing teacher self-reporting of practices through the
interview and observing teaching practices will allow for triangulation. Focus groups were
considered in order to reach more teachers; however, because the ability to enact EMTPs may be
a sensitive topic, I chose to maintain a one-on-one interview with each participant in hopes that
each teacher would freely share their thoughts.
Interview Protocol Design
Several resources were utilized to design the interview protocol. Patton (2002), Merriam
and Tisdell (2016), as well as Robinson and Leonard (2019) served as a framework for question
and overall interview protocol design. To better understand the indicated motivational
influences, the Self-Efficacy for Teaching Mathematics Instrument (SETMI), which measures
self-efficacy related to pedagogy in mathematics and teaching mathematics content, was used as
a foundation for interview questions (Enochs, et al., 2000). Additionally, ideas from the
Mathematics Quality of Instruction (MQI) framework were used to design questions which gain

64
understanding of the content knowledge influence (Learning Mathematics for Teaching/Hill,
2014). Math units for Grades 3–5 were examined to determine which concepts teachers were
teaching in order to design the questions about content and progressions (See Appendix A:
Interview Protocol for Teachers). For the remaining factual, conceptual, and procedural
influences, indicators on the observation checklist were used to design interview questions. The
development of the observation checklist, and the sources used, are described in the following
section.
Observation Checklist Design
An adapted version of the mathematical teaching practices continuum (NCSM, 2019) was
used to capture observational data (See Appendix B: Observational Tool). Ideas from the
Mathematical Quality of Instruction (MQI) framework (Learning Mathematics for
Teaching/Heather Hill, 2014), the culturally relevant, cognitively demanding task framework
(Mathews et al., 2013), and the equitable mathematics teaching practices were incorporated
(Bartell et al., 2017). In order to ensure that the tool captured what was needed, I first compiled a
list of the indicators for the knowledge influences. The mathematical teaching practices
continuum was then compared to these indicators. Indicators found on the continuum not
specifically being studied were removed. These included establishing mathematical goals to
focus learning and building procedural fluency from conceptual understanding. Indicators that
were not found in the continuum, but which need to be studied, were added. These included tasks
that build conceptual knowledge or application, tasks that promote understanding, critiquing, and
changing the world with mathematics, ensuring that tasks are authentic and build on children’s
experiences and funds of knowledge, and attending explicitly to race, culture, and other
identifying characteristics. However, this very detailed tool was too long to be used effectively.

65
As such, I looked for themes across different sections and minimized the amount of observable
teaching behaviors while still capturing what was required to learn about the knowledge
influences. This left the tool with five areas for observation including tasks, discourse and norms
of participation, meeting students’ needs, content, and attending to race, culture, and other
identifying characteristics (See Appendix B: Observation Tool).
Data Collection
Following University of Southern California Institutional Review Board (IRB) approval
and recruitment of participants, data was collected through semi-structured interviews and
classroom observations.
Interviews
Semi-structured interviews were conducted with each of the six selected participants in
person before observations. This gave me an opportunity to build rapport with participants before
observing them. A proxy was not needed as I am an instructional coach for grades preschool–
Grade 1 and I do not frequently work with teachers in Grades 3–5. Additionally, I do not
supervise any of the participants.
An interview protocol was used to guide the general direction of the interview. The
interview protocol was rehearsed so I knew how the interview should flow and where to record
information provided by participants for a different section than what was being presented at the
time. Practice prompts were rehearsed, and a copy of the practice prompts were available to
ensure closed questions were not asked following participants’ responses.
Times for the interviews were decided upon according to the convenience of the
participant. Each interview took approximately 30 minutes to one hour to complete and was
conducted by me in a quiet space without interruptions, in a space that was decided upon by the

66
participant. The interview was recorded using a voice recording app, Otter.io. An iPad was used
as a secondary device for recording in case the first device was faulty, or quality was poor. Notes
were taken with paper and pencil to avoid distraction or technological difficulties. A digital
watch was used ensure that the interview was progressing appropriately. I wanted to honor what
each participant was saying and needed to move the interview along. I needed to have a general
conception of how long each section should take and ensure that we are remaining within time
limits in order to gather all relevant data. Audio recordings were automatically turned into
transcripts using Otter.io. After transcripts were uploaded, they were compared to the audio
recording for accuracy. This information was all shared with participants prior to the start of the
first interview.
Observations
Two 45–60-minute classroom observations were conducted with each of the six
participants. As I do not coach teachers in Grades 3–5 nor supervise any of the teachers, I was
able to conduct the observations. The observations took place during the normal math block and
the times and days were chosen by the participant within a two-week window. The reason for the
two-week window is to keep the observations within a single unit so that the content does not
change. Information about content knowledge will be extrapolated from the observations by
looking for evidence that the teacher is asking questions that pushes children to understand the
math content and giving explanations that are mathematically correct.
An attempt was made to arrange the observations during a time where all teachers were
teaching a similar content, despite their grade level. While this was possible for some, not all
participants were teaching a similar idea. This did not detract from the findings.

67
To ensure authenticity within the classroom, I provided clarity on my role as a researcher
prior to observing. I explained to the teachers that I am strictly an observer and while it was okay
to explain to children that I was there learning about mathematics teaching, I would not interact.
The mathematical teaching practices continuum (NCSM, 2019) with adaptations as
described above was utilized to capture data. Consideration was given as to whether the
observation tool would be shared with teachers prior to the observations. The decision was made
to not share the protocol with teachers ahead of time so that they do not attempt to change their
practices according to the observation tool, which would skew the data collected. Additionally,
to capture accurate data during the observation, it was discussed with the teachers ahead of time
that I wanted to capture the true reality of classroom instruction in mathematics to understand the
teaching that is taking place in order to make recommendations to the school regarding how to
best support teachers in teaching mathematics. Photos without children were be taken to capture
the room arrangement.
As the adapted continuum was still robust after minimizing the number of observable
teaching behaviors, I took steps to ensure that I was able to collect data accurately. To begin, I
used the tool in several classrooms prior to official data collection to become familiar with it. I
reflected on my ability to capture important details and adjusted as needed. I also sought
permission from the teachers to video record during the observation so that I was able to go back
and code the teacher and children's contributions. Thought was given to whether video recording
would give biased data as the children and teachers might interact differently with the presence
of an observer and a video recording device. However, the cons of possible biased data
outweighed the pros of being able review and analyze the video for further information.

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Data Analysis
Data analysis began during the data collection phase as reflective notes were written after
each observation and interview to document ideas, questions, connections, and possible codes as
they emerged. Both a priori and inductive codes were used. A priori codes were developed
utilizing the conceptual framework and research questions. Further codes were developed as they
emerge from the survey transcripts and video recordings of observations. Excel spreadsheets
were used to aggregate common codes to find patterns in knowledge and motivation influences
as well as facilitating and inhibiting organizational influences that explain the possibilities for the
ability or inability to enact EMTPs. Inferential or descriptive claims were made upon the
convergence of data.
Trustworthiness of Data
The credibility and trustworthiness of a study is of utmost importance. Researchers need
to consider how they might be wrong or how worldviews or biases might impact the research
process. These questions impact all parts of the research process from identifying the problem
and writing research questions, to deciding how to collect, analyze, and report data. However,
because qualitative inquiry relies on human interpretation and meaning making, and the
researcher is the primary instrument, it can never be completely unbiased or trustworthy
(Merriam & Tisdell, 2016). As Maxwell (2013) noted,
Validity is never something that can be proved or taken for granted. Validity is also
relative: it has to be assessed in relationship to the purposes and circumstances of the
research, rather than being a context-independent property of methods or conclusions.
(pg. 121)

69
Fortunately, there are ways to increase the trustworthiness of the process of your study
and credibility of your findings. Purpose is important to consider when determining what
credibility and trustworthiness checks you are engaging in for your study. Rather than treating
checks as a checklist, one must consider the most plausible credibility threats and plan
accordingly (Maxwell, 2013).
In my study, there were several things to consider in order to increase credibility and
trustworthiness. To begin, I engaged in what Erickson (1973) termed “disciplined subjectivity”
to ensure that I understood my values, views, and biases and how they may shape or influence
my study through reflexive activities. These included journaling and memo writing (Atkins &
Duckworth, 2019; Maxwell, 2013). Due to my strong history and known feelings and
positionality with this topic growing up, as a mother, and as a teacher and coach who sees
inequitable opportunities for deep mathematics learning, I kept notes on interactions that I had
with others to ensure that my biases were not encroaching on my research. As I completed my
literature review, I was aware of confirmation bias and tuned into evidence of alternative
possibilities intentionally rather than selecting literature or data that fit into my existing tacit
theories (Ravitch & Carl, 2012) and ignoring literature and ideas that do not fit with my
preconceived notions (Maxwell, 2013). I remained aware of this confirmation bias as I
conducted my interviews and observations to ensure I was collecting the true perspectives of
teachers. These steps demonstrate transparency to my participants so that I can be seen, and the
process can be seen, as trustworthy, and the findings valid. To ensure credibility, during
observations I collected thick, rich descriptions to represent the situation as accurately as
possible.

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Limitations and Delimitations
There are several limitations and delimitations to this study. One limitation was the short
timeframe for the study. As data was collected over a short time period, only a limited number of
observations and one mathematical content domain was assessed. The time frame also limited
the number of participants and schools to be studied as there was not enough time to conduct
more interviews and observations. A longer time period would have allowed for a more in-depth
study, revealing themes over time and could have potentially been informative for other schools.
However, while claims cannot be made beyond the school, it has been determined that the
sample size is appropriate for the study and thus the findings were useful for the stakeholders
within the study site.
Further limitations resulted from the data collection methods. While the observational
tool was synthesized into themes to reduce the number of observable teaching behaviors, so the
tool was manageable, there are still several observable teaching behaviors. It was challenging to
capture data for all observable teaching behaviors in the observational tool. The use of video and
extensive notes was used to overcome this limitation. However, video recording posed a further
limitation in that it may disrupt the authentic environment, resulting in biased data. Assessing
teacher bias through observations was difficult as well. While there might be certain indications
of bias such as refraining from calling on a child, there is no way to confidently say that this
behavior is a result of bias.
There were limitations within the interview process as well. While I am not an
instructional coach for Grades 3–5 and do not supervise any of the teachers, most teachers know
that I am passionate about mathematics. While I continually reminded teachers of my strict
research role to learn more about mathematics teaching to make improvements, there was still a

71
chance that teachers were cautious with what they shared. Limiting opinions may also have come
as a result of the culture of the school where some teachers may not feel safe or confident in
acknowledging a lack of knowledge or skill or critiquing the school.
An additional limitation may have been biased literature. While due diligence was used to
locate literature from various perspectives to inform the conceptual framework, there is always a
possibility that some literature was missed unintentionally.
Delimitations result from the selection criteria and process. Grades 3–5 teachers were
only included in this study. This was specifically decided because the developmental level of
children in Preschool–Grade 2 is different from children in Grades 3–5, and thus children in
Preschool–Grade 2 may need more structured teaching approaches. Additionally, the study did
not include teachers who have been at the school less than 3 years as I wanted teachers to be
familiar with the school. The study could have benefitted from the unique perspectives of newer
teachers who have not been acculturated into the school. Further, non-classroom teachers that
teach mathematics such as learning support teachers were not part of the study. This was decided
in order to limit the observations and interviews to those who conduct initial, whole class
instruction where children are grouped heterogeneously as is recommended in the literature for
equitable opportunities to deep mathematics learning.
Another delimitation posed by the selection process was the voluntary nature. As the
study is voluntary and those who respond first will be selected, this could provide biased results
as those who volunteer may feel especially passionate about issues or more confident in their
abilities, thus reducing the representation across teachers Grades 3–5.
School culture and leadership play a significant role in the ability to enact EMTPs. As
these two influences are beyond the scope of the study, it was difficult to say to what degree the

72
culture or leadership did not facilitate or inhibit enactment of EMTPs rather than the identified
influences in the study. While not specifically asked, both leadership and culture were mentioned
by several participants. Further, one aspect of enacting EMTPs was supporting the development
of a positive mathematics identity for all children. While teaching moves and their possible
influences on children’s math identity can be considered, for the purposes of this study, we could
not measure the impact of the enactment of the EMTPs on children’s math identities because we
are only engaging teachers as participants. Furthermore, as we were not engaging children or
their formal assessments in the study, we do not know if the teaching practices resulted in deep
mathematics learning. We can make predictions based on what we observe during the teaching
episodes but cannot make specific conclusions.
Ethics
It is of utmost priority to keep the participants' interests, needs, as well as psychological
and physical safety centered (Lochmiller & Lester, 2017). There are several ways participants
could be harmed, but there are also ways in which participants can be protected from such harm.
To begin, participants may feel obligated to participate in the study because I am a math
instructional coach. However, essential to note is that I have no supervision duties in relation to
the participants and do not work with teachers in Grades 3–5 often as my primary role is to
support teachers in Preschool through Grade 1. One way I offered protection from feeling
obligated to participate is through obtaining consent from participants (Lochmiller & Lester,
2017). Prior to the study, participants understand the purpose of the study, why they’ve been
invited to participate, that their participation is voluntary, and that they can refrain from
answering any questions they choose. Additionally, I was very upfront with participants that I
hold biases about the types of mathematics teaching that ensures equitable opportunities for all

73
and have reflected on how to ensure my biases will not impact my data collection, analysis, or
the way my findings will be presented. I let them know that I am strictly interested in learning
about how teachers are teaching mathematics at the school and how they feel supported to enact
EMTPs so that the organization can make improvements. I informed them that the nature of my
research is not evaluative, and I will not be making any judgments about how they are
performing as a teacher. Before the interview, I reminded them of these conditions and asked for
any questions they might have had about participation.
Participants may also be harmed if information is shared with others, especially
administration, about their teaching practices if they do not align to what the school expects.
Additionally, participants may be harmed if they speak out against the school and that is shared
with the organization. As such, confidentiality was important to ensure that respondents were not
harmed. As respondents are answering questions about their beliefs and teaching practices, some
of which might not align to the direction of the school, I ensured respondents understood that I
will make anonymous anything that I can. This was done by making quotes and observations
anonymous by removing identifying information from the findings. I asked to use specific quotes
before doing so. I let participants know that all data will be stored in a password protected
computer and will be destroyed after 3 years. As teachers know that instructional coaches meet
with principals to discuss instruction, I also assured participants that any data collected during
interviews and observations will not be shared with the administration, so participants did not
have a fear of getting into trouble because of their mathematics beliefs or teaching practices or
their perspectives on how the school supports them or not.
Robinson and Leonard (2019) take a slightly different approach to ethics when they
suggest that researchers reduce the burden as much as possible on participants. One way I

74
accomplished this is by being focused in the interview by only asking questions related
specifically to answering the research questions. As a learner in general, I know that I often like
to get as much information as possible because it is fascinating to me. Prior to engaging in the
interview, I reflected on how I might take different paths based on what teachers say in the
interview and identified places where the potential to go off topic exists and came up with ways
to avoid this.
An additional ethical concern involved the roles a researcher may take according to
Glesne (2011). One role that I needed to avoid is the exploiter. As an instructional coach who has
studied mathematics, I am aware that I have biases about what EMTPs should look like. Part of
my role as instructional coach is to find places where teachers might want to improve, which
oftentimes places the onus on the teachers for improvement. While the study highlighted
practices that contributed to inequitable opportunities in mathematics, and as instructional coach
it is my role to support teachers to improve in these areas, the study needed to address the
systemic causes (Patel, 2015) rather than placing the onus of a lack of knowledge and motivation
for enactment of EMTPs on the teachers.
Positionality
Villaverde (2008) defined positionality as “how one is situated through the intersection of
power and the politics of gender, race, class, sexuality, ethnicity, culture, language, and other
social factors” (p. 10). Understanding your positionality as a researcher is important as it impacts
all aspects of the research process, especially the participants. It is important to interrogate your
positionality to ensure that you select research strategies that minimize the impact of your
positionality. One specific way I accomplished this was conducting a bracketing interview in
which a colleague interviewed me to share my experiences and feelings about mathematics and

75
equitable opportunities for learning (Lochmiller & Lester, 2017). This was done prior to data
collection.
My life experiences have contributed considerably to my interest in this topic and the
potential for bias exists. During this study, I held those experiences close while understanding
the impact they may have on the entire research process. I engaged in what Erickson (1973)
termed “disciplined subjectivity” to ensure that I understood my values, views, and biases and
how they may shape or influence my study through reflexive activities such as journaling and
memo writing (Atkins & Duckworth, 2019; Maxwell, 2013). Additionally, as I conducted my
observations and interviews, I was aware of confirmation bias and tuned into evidence of
alternative possibilities rather than selecting data that fits into my existing tacit theories (Ravitch
& Carl, 2012) and ignoring ideas that do not fit with my preconceived notions (Maxwell, 2013).
Growing up, I did not have a positive mathematics experience. I believed math was about
replicating what the teacher taught me and getting the right answers quickly. I did not feel
successful in mathematics until I took a course at university that helped pre-service teachers
understand and prove the mathematics they were going to be teaching their children. This
negative childhood experience could have contributed to how I wrote questions and the way I
interpreted teacher responses and observations. For example, I needed to be careful not to ask
leading questions such as “How do your teaching practices prevent some children, especially
those who struggle or are from marginalized backgrounds, from having opportunities to engage
in rich mathematics?” nor listen or look for specific responses that reflect my experience such as
the teacher teaching in a traditional way.
I have been a teacher and an instructional coach for several years in the organization that
is being studied. I have also been involved with two math reviews, one as a participant and one

76
as a co-facilitator. Additionally, I am the math team coordinator K–5. While my position as
instructional coach is not an official position of power, I could be perceived as holding positional
power by teachers. As mentioned, however, I do not hold any supervisory duties nor work often
with the participants in this study. Additionally, due to a lack of a consistent coaching culture,
many teachers might perceive me as an expert who will show them the ‘correct’ way to do
something. Furthermore, many teachers, despite not working directly with them, know that I am
passionate about equitable mathematics and actively question some of the inequitable practices
that are occurring at our school. This has great implications for my research. For example,
teachers may not disclose practices that they know I do not believe to be equitable. Additionally,
teachers may try to answer questions in a way that they may think I want to hear. To ensure this
did not interfere with my research, I needed to minimize the potential impact on participants and
other aspects of the research process. Within the interview, I was very careful to compose
questions that were objective so that teachers do not perceive the interview as collecting data to
support what I might think is right or wrong. An additional way I attempted to ensure this does
not interfere with my research was through offering reminders to participants that I was strictly
conducting the study to learn about mathematics teaching and the support teachers receive to
make improvements and that my research was not evaluative. Furthermore, I needed to be clear
and transparent with my participants about my biases and the strategies I planned to use to
understand and protect against these biases. One such way I did this was by keep notes on
interactions that I have with others to ensure that my biases are not encroaching on my research.
The positionality of myself as a friend had the potential to impact my research as well
(Glesne, 2011). It was necessary to ensure that I did not exclude data that could highlight a friend
as not holding the knowledge or motivation to enact equitable teaching practices in mathematics.

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To do this, I journaled each of my interactions and reflected on the data collected to be sure that I
was being objective. Additionally, I needed to ensure that teachers were speaking to me as a
researcher rather than a friend. To ensure this was the case, I reminded them throughout the
interview that I would be using what they say in my research. When someone mentioned what I
thought to be a personal comment, I addressed this by asking them if I could share it.

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Chapter Four: Results and Findings
This study aimed to conduct an adapted gap analysis to examine the knowledge,
motivational, and organizational influences impacting the successful enactment of equitable
mathematics teaching practices (EMTPs) of grades 3–5 teachers in the elementary school. This
study was informed by an adaptation of the Clark and Estes (2008) gap analysis model, a model
which describes the process of human performance. The conceptual framework (see Figure 3)
utilized in this improvement study outlined two declarative knowledge influences, one
procedural influence, two motivational influences, and three organizational influences that have
the potential to impact the enactment of EMTPs. The aligned questions guiding this study
included:
1. What is the current status of teachers’ knowledge and motivation related to enacting
equitable mathematics teaching practices, or those teaching practices that provide
access to deep mathematics learning for all learners?
2. How do organizational factors influence teachers’ capacity to enact equitable
mathematics teaching practices?
To validate the assumed influences, two qualitative data sources were utilized,
specifically, semi-structured interviews which allowed for flexibility (Lochmiller & Lester,
2017) and classroom observations to understand the knowledge, motivational, and organizational
influences impacting the enactment of equitable mathematical teaching practices. Each
participant engaged in two interviews and two observations. Interviews were conducted prior to
the observations so the researcher could gain rapport with the participant and so understood the
mathematical content being engaged with during the observations could be understood.
Interviews were audio recorded and transcribed using Otter. Observations were video recorded

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and analyzed using the observation tool developed for this study (See Appendix B) which was
not shared with participants before the observations. Transcripts and video footage were then
coded using a priori codes based on the KMO influences as well as inductive codes from
emerging themes. Results from the analysis of transcripts and video will be organized by the
assumed influences and inferred to be either an asset or a need based on the triangulated data.
Participating Stakeholders
The primary stakeholders in this study were a purposeful sample of elementary school
teachers who teach Grades 3–5. This group is directly responsible for enacting EMTPs that result
in deep mathematical learning for all. In total, six teachers agreed to participate in interviews and
classroom observations for the study. Two teachers per each of the grade levels represented
participated. Five participants were female while only one participant was male. While an
attempt was made to choose two teachers who utilized more traditional teaching methods, only
one participant who volunteered fit these criteria. Four participants have taught grades 3–5 for
the duration of their careers while two have taught primary grades with grade 3 being the highest
grade level taught. The average teaching span was approximately 10 years. One participant has
also taught in gifted and learning support programs. All participants have taught at Green View
River International School (GVRIS) for a minimum of 3 years. Only two of six participants had
a positive experience with learning mathematics in elementary school while the other four had
negative or neutral experiences.
Determination of Assets and Needs
Determination of assets and needs was made based on analysis of the six interviews and
six observations. Copur-Gencturk and Thacker (2021) found that self-reports do not often match
direct assessments, thus direct observational assessments were utilized in conjunction interviews

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to triangulate data. Furthermore, to ensure a quality, triangulated data set, each participant
engaged in two observations so trends could be identified. Finally, in designing the interview
protocol, multiple questions were utilized to elicit multiple data points for a particular influence.
For each influence, criteria were set for determining whether data supported the
indication of an asset or need. For some influences, the determination as need or asset was clear
due to the ability to collect numerical data, for example, to determine whether teachers know the
goals of mathematics or mathematics teaching. For influences with such clear determination
criteria, four of six participants had to have validated the influence for it to become an asset. For
those with many indicators, or those with a range within an indicator, the researcher considered
the data holistically. One such indicator involved engaging students in discourse, a subset of the
procedural knowledge influence of being able to successfully enact EMTPs. Discourse is further
divided into several aspects such as pressing a child to explain their thinking and orienting
children’s’ thinking towards others to develop a shared understanding. Some participants
attempted to engage children in various aspects of discourse, albeit with a range of success.
Hence, a holistic approach across subsets and aspects had to be considered universally. A final
approach to determining a need or asset involved the comparison of self-reports to direct
assessment and examining the alignment or misalignment between the two measures. When
alignment was present, the influence supported the consideration of the influence as an asset. On
the other hand, when misalignment was present, an influence was considered a need. The
findings from this study are organized into three categories: knowledge, motivational, and
organizational influences (KMO). For each influence, the data examined indicated the influence
to be an asset or need. The specific KMO categories assessed were as follows:
• Knowledge influences: factual, conceptual, and procedural

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• Motivation influences: teacher efficacy and collective efficacy
• Organization influences: cultural settings
Results and Findings for Knowledge Causes
Understanding the types of knowledge required to meet goals is necessary to meet them
(Rueda, 2011). Four specific types of knowledge have been identified by Anderson and
Krathwohl (2001): factual, conceptual, procedural and metacognitive. For this study, factual,
conceptual, and procedural knowledge have been evaluated.
Factual and Conceptual Knowledge
For this study, three factual and conceptual knowledge influences were examined.
Factual knowledge entails basic, concrete knowledge (Anderson and Krathwohl, 2001).
Conceptual knowledge refers to more complex ideas or theories.
Influence 1
Teachers know and understand goals of mathematics and mathematics teaching.
Interview Findings. Responses to four interview questions addressing the goals of
mathematics and mathematics teaching were utilized to determine whether this knowledge
influence was an asset or need. The identified goals of mathematics and mathematics teaching
were to see beauty, wonder, and joy in mathematics, to position children as knowers and doers of
mathematics, and to be used as a tool to understand, critique, and change the world. Findings
revealed that all participants could articulate that one goal of mathematics teaching was to
position children as knowers and doers of mathematics. This includes such ideas as supporting
problem-solving, sense-making, flexibility, and application. One participant stressed the
importance of positioning children as capable by ensuring children can explain and justify
reasoning:

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We need to teach kids from early on how to justify their answers and explain their
thinking so that when they get to a point where it's not possible just to jump to the
answer, they can't do it mentally anymore, that explaining of their thinking and justifying
their reasoning has become automatic.
Two of the six participants, or 34%, stated that while they believed positioning children as
capable mathematicians is a goal of mathematics, the education system at our school did not
support this. Instead, they reported that the focus was on knowing how to accurately complete
procedures and take tests. Two of six participants understood that mathematics is a tool to
understand the world. Participant 5 stated, “All fits under the umbrella of just being able to
function in a world that relies on numbers and quantities and spatial reasoning.” The goal of
seeing the beauty, wonder, and joy of mathematics was not significantly evident. Furthermore,
no participants named using mathematics as a tool for changing and critiquing the world as a
goal of mathematics.
Observation. Five of six participants observed engaged in lessons that supported the goal
of mathematics and mathematics teaching as positioning children as knowers and doers of
mathematics with varying levels of success. In one classroom, there was a sense of joy as
children worked together to solve problems while in the other classrooms, children were working
to solve the problem out of what appeared to be compliance rather than joy. Understanding the
beauty and wonder of mathematics as well as understanding, critiquing, or changing the world
were not observed.
Summary. The assumed declarative knowledge influence, knowing the goals of
mathematics and mathematics teaching, was determined to be a need as a result of interview and
observational data. Positioning children as knowers and doers of mathematics, or attempting to

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do so, was strongly evident in both interviews and observations. Understanding the world was
noted to be a goal in the interviews; however, this was not observed. No evidence was found for
the remaining goals of mathematics. As a result of these findings, knowing the goals of
mathematics and mathematics teaching is considered a need.
Influence 2
Teachers know and understand what EMTPs are.
Interview Findings. Knowledge of the understanding of what EMTPs are was examined
through eight interview questions. Findings varied. When asked to describe their teaching
approaches not specifically related to equity, all participants were able to name several teaching
practices that support equity in mathematics classrooms. One participant asserted that they liked
to use collaborative inquiries where children were engaged in discourse to make meaning and
where the teacher is attending to student thinking as illustrated by the following comment:
Then we move into some sort of like open-ended problem that can be solved in multiple
ways. When we do that, kids are most often collaborating. That’s where I kind of like
diverge. It’s like we have all these paths to go. I’m kind of anticipating the different ways
that it would go and I’ve got to be able to shift gears depending upon what happens.
While it was found that all teachers were able to describe some practices that are considered
equitable according to this researcher’s literature review, when asked specifically what EMTPs
are, teachers were mostly unable to describe which practices support equity in a mathematics
classroom or why. Five of six participants were able to indicate that equity in a math classroom
means that all children should have access, feel a sense of belonging and validation through
differentiation, collaboration, discourse, and ability to solve problems in a variety of ways.

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Beyond this, however, teachers were largely unable to name other specific practices that support
equity in mathematics classrooms.
Absent in the interview findings was knowledge of practices related to culturally
responsive teaching including using authentic problems beyond using children’s names in the
contexts, building on funds of knowledge, and attending explicitly to race and culture. In one
case, a participant equated equity issues in the math classrooms to socio-economic status, race,
and trauma when they commented:
It’s a lot different here than it would be in the States because largely our kids are all at a
reasonable SES [socio-economic status] and so we are not having a bunch of traumatized
kids come in, kids come in who need something besides math and I know I have a lot of
blindness, but I don't see color as an issue.
Observation. Observations were not conducted for this influence.
Summary. Evidence presented in interview results in the preceding section indicated that
the assumed declarative knowledge influence, teachers need to know and understand what
EMTPs are, is a need. While most teachers referenced the use of some EMTPs, most participants
were unable to identify or explain specific teaching practices that support equity in mathematics.
Influence 3
Teachers know and understand mathematical content knowledge and progressions.
Interview Findings. Interview questions revealed various levels of content and
progression understanding and how progressions could be used to support student learning. Five
of six participants were mostly able to explain the big ideas of the mathematics they were
teaching, although imprecise language was predominantly used to do so. Moreover, four of six
participants demonstrated underdeveloped ideas about at least some of the content they were

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teaching or relied on shortcuts or tricks. Participant 4, when asked about how having an
understanding of unit fractions helped children understand how to add, subtract, multiply, or
divide with fractions, responded, “When you understand the unit fraction you understand why
you're adding numerators and denominators. When you have fractions, some of the values
change and some do not.” This suggests that while the participant maintained some
understanding that the denominators do not change when adding fractions, they were possibly
unclear, or at least unable to articulate at the time, that the reason the denominators do not
change, and the numerators do change, is a result of adding like-sized pieces. In addition to this,
only half of participants were able to articulate the content learning progression. Nevertheless, all
participants could articulate various strategies and visual representations children might use
when solving problems related to the content area. Despite this strength, confusion between
strategies and models was demonstrated among two participants. An additional positive finding
in the data related to content was the ability to articulate common misconceptions and/or
challenges, which was successfully expressed by all participants. However, only one teacher
used content progressions to support students with challenges or misconceptions. Participant 5
demonstrated using a simpler idea within the learning progression when they stated,
In relation to a fraction multiplied by a whole number, they would probably, if they
either know how to relate it to multiplication in general, and how it fits in with repeated
addition. So again, they either come knowing that because someone's told them just times
that. “Oh, and remember, this is actually 6 over 1, we just don't write one,” the rule. So,
they either know just to do it. So, I would prompt them like, “What would you do if you
were multiplying? Or if you had, you know, if, if you just had three? What do you know
about multiplication?”

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While utilizing the content progression was not demonstrated, the remaining five participants
reported relying on the concrete-representational-abstract progression to support children with
challenges or misconceptions by asking them to model or visualize the problem.
Observation. Observations of the use of correct content knowledge and progressions
were varied. Three classrooms utilized correct content for the duration of the two observations in
various ways including anticipating what misconceptions children might have, what strategies
they might use, or what might be challenging as well as engaging in conversations during small
group work. In an unsuccessful attempt to utilize content correctly to correct a misconception,
Participant 4 explained:
Okay, so, what I think I’m hearing you say is that, that when you see a big number is
how you could break it up in a more [inaudible] and the understanding that you’re not
multiplying 3 times 6, but you are saying 3 times 60, right?
Participant 4 then abandoned the attempt after a cursory explanation. Other misconceptions were
left uncorrected completely. A further point is that while Participant 1 engaged with the content
correctly, it was more procedural than conceptual. Utilizing the content progressions for
extension was not observed.
Summary. While interviews revealed that most teachers have basic content knowledge,
know strategies and visual representations, as well as common challenges and misconceptions,
most participants were unable to articulate the content progressions or how they could use
progressions to support learners. Observations revealed that only half of participants utilized
correct content knowledge for the duration of their two classroom observations. As such,
evidence is presented which supports knowing and understanding content knowledge and

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progressions as a need. It should be noted that four participants suggested the importance of
learning content progressions and how to utilize them in support of meeting students’ needs.
Procedural Knowledge
According to Anderson and Krathwohl (2001), procedural knowledge describes the
knowledge of how to do something. One procedural knowledge influence was examined for this
study consisting of five categories as described in the following section.
Influence 1
Teachers know how to enact EMTPs.
Interview Findings. No interview was conducted for this influence.
Observation. The successful enactment of EMTPs was measured through observations
using an observation tool with the following categories: 3-part lesson framework, tasks,
discourse and norms of participation, meetings student’s needs, content, and attending to race,
culture, and other identifying characteristics. In the sections that follow, each category will
highlight the related findings.
Findings related to a 3-part lesson and tasks were varied, with some strengths and areas
of need. Observations revealed that five of six participants structured their lesson with a 3-part
framework, with varying degrees of success. To illustrate, two-thirds of participants utilized
tasks that were appropriately challenging with multiple entry points and solution paths. However,
when launching the task, only two participants invited speculation. One asked, “What do you
notice?” The other, Participant 2, allowed children to ask questions about the mathematics in the
task. Ensuring students make a plan for their work, as well as ensuring the expectations are clear,
which other indicators of an effective launch, were only observed in one classroom.

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Utilizing tasks that are authentic, and thereby help children to connect mathematics to
their lives, is another mathematics teaching practice which falls under this influence and
category. Half of participants made use of such tasks. In Participant 2’s classroom, an inauthentic
context was revealed to children. Children responded by changing the context, so it made more
sense to them. While the participant did not present the authentic context, they allowed the
change, an indication that they want children to make sense and value their voices. Tasks that
provide opportunities for children to understand, critique, and change inequities or social justice
issues within the world with mathematics were not observed.
To ensure children feel supported and valued in the mathematics classroom, discourse
and norms of participation are required. Establishing norms of participation where all are
expected to participate, and children are supportive towards each other, was found in all
classrooms to varying degrees. One example was modeled by Participant 5 who engaged in
similar lines of questioning with all children to ensure they were working towards the learning
goal, as evidenced in the following exchange between teacher and child. Participant 5 asked the
child, “What was your approach? What’s your first move?” In response, the child reasoned
through the problem. Participant 5 pushed further, asking, “What’s that going to look like?” The
child worked through the solution and exclaimed, “It works!” This type of exchange is indicative
of having high expectations for all students.
Five of six classrooms were found to be safe spaces for children to ask for and receive
help from their peers as well as share ideas. In one classroom where this safety and positive
interaction was not observed, children seemed to feel self-conscious about their skill level and
willingness to ask for help. Child 1 asked in a frustrated tone, “How do you know where to put
this stuff?” Child 2 asked in response, “Oh, do you need help?” Child 1 responded, “No.” It can

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clearly be seen that the student did not feel safe to receive help from their peers. One important
observation occurred only in one classroom but could have the potential to impact how students
feel about themselves as mathematicians. At GVRIS, there is a math lab, a class in which those
who score well on assessments can attend. During Observation 2 of Participant 3’s class, the
children who go to math lab were not present. It was noted that while the willingness to ask for
and give help was present during Observation 1, during Observation 2, the children seemed less
willing to persevere and take risks in helping one another. This finding may indicate that when
children are tracked, they feel less successful when alone in the groups considered ‘lower.’
A classroom culture which fosters sense-making was found in five of six classrooms.
This was evidenced through the teacher questions. In two of five classrooms, observations
revealed children prompting their own sense-making through questioning when asking questions
such as “Do you think we should start with this?” or “How are we going to solve this?” Fostering
curiosity, which promotes joy and wonder, however, was largely absent.
Discourse, a tool to support students maintaining the authority of knowledge and to create
shared meaning, revealed various strengths and needs. Only one of six teachers was observed
purposefully prompting children to talk about each other’s explanations. Participant 4
encouraged this by asking, “Can anyone restate what [child’s name] said? Does anyone have
anything to add on?” While this prompting was present, it was at a minimal level. Most
prompting in this classroom was to check for agreement or disagreement without discussion or to
ask for other ways of thinking.
One important aspect of discourse is pushing students beyond what they are doing to
explain why something works. In four of six classrooms, participants consistently pushed
children to move beyond the ‘how’ to do something to the ‘why’ it works. Questions asked by

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Participant 4 such as, “Why are these strategies the same or different?” or “Why does this work?
How would we prove this?” or “Why? Where do these numbers come from?" from Participant 2
indicate this high-level expectation. Participant 6 reminded children that the ‘why’ was important
by stating, “The important part is the ‘because’ at the end. Why do you think that?” One teacher
pushed students to explain why things work inconsistently. On the day where the children who
go to math lab were present, the teacher asked mostly open questions such as, “Is it? Because?”
However, on day two of the observation when the children who attend math lab were not present,
this practice was used less consistently. Instead, some of the questions asked were closed such
as, “Do you think it will be faster or slower? Did it start and end at the same time?” In one
classroom, the participant used some questions to appear to push thinking about the ‘why’ but
then either gave feedback and affirmation on a single way to do something or did not give wait
time after the question and moved on, failing to return to the question. In one instance, the
teacher posted a slide that said, “Explain” and there was a collective groan from the children.
The participant replied by skipping the slide.
Pushing children to argue the validity of a mathematical statement or solution through
reasoning, another aspect of discourse, was present in two of six classrooms. These two teachers
pushed children to use reasoning to prove their thinking through advancing questions. In
Participant 5’s classroom, one child stated, “You always put the one on the bottom.” The teacher
replied, “Why? Always? When you say it works, what do you mean by that?” Participant 2 posed
questions such as, “What made you do that?”
Critiquing reasoning of other children pushes students further into building collective
meaning. This was absent in observational data. While there were attempts made by some
participants to have children revoice the ideas of others to check for understanding, or to ask if

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other children agree or disagree with the child sharing, there were no observations of teachers
asking children to critique the reasoning of their peers.
Similarly, there was little evidence of teachers pushing children to build on strategies or
thinking of others. This practice was only evident in two of six classrooms. In Participant 4’s
classroom after a gallery walk to view the work of others, the teacher asked the children to
consider the thinking of others by asking, “Are you thinking something different? Would you
change? Are there other ways?” Furthermore, in the summary portion of this participant's lesson,
there was no evidence of children building on the ideas of others. Instead, the interactions were
between the participant and child while the other children listened in without the obvious
expectation that the reason for sharing was for others to consider how they might build on to the
thinking or compare their thinking. Participant 5 attempted to push children to consider how their
thinking compared to others’ thinking when they stated, “Some people are saying … while
others are saying they solved it this way. How is your thinking similar or different?”
To support collective meaning making, Participants 2 and 4 intentionally anticipated,
monitored, and selected solutions for whole group discourse to move towards the teaching goal.
However, when it came to sequencing these ideas for a certain purpose, the purpose seemed to be
unclear for both teachers and children, as illustrated by the lack of engagement and discussion.
Despite partly successful attempts at discourse, the authority seemed to reside in the
thinking of the children rather than the teacher in all but one classroom and one day of another
classroom. As noted previously, in Participant 3’s classroom during Observation 2, the teacher
took on more of the authority when the children attending Math Lab were not present. In the
other four classrooms, authority of the child was illustrated through the questions asked by the
teacher and the consistent expectations that all children are responsible for their thinking.

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To maintain equity in a math classroom, one must meet the children where they are. A
majority of classrooms utilized formative assessment data to understand the needs of students. In
response, observations revealed that in four of six classrooms, participants provided scaffolds for
children to access the mathematics without lowering the cognitive demand through assessing and
advancing questions. The one exception to these four classrooms was in Participant 3’s
classroom on Day 2 when the math lab children were not present as described previously. In
some cases, Participant 3 lowered the cognitive demand for some struggling children by
engaging in closed questioning and direct teaching. Participant 4 engaged in this line of
questioning to support a child without lowering the cognitive demand who was struggling to get
started, “So, this represents. … How are you going to do that? What else might you do? What
would that look like in terms of this context? I’ll come back.”
The use of the concrete-representational-abstract (CRA) progression to support children’s
thinking was evident in one classroom as a scaffold; however, concrete tools were only made
available when the teacher deemed it to be necessary. In Participants 3 and 6’s classrooms, all
children were given the same concrete objects as was suggested by the resource. As such, the use
of CRA for scaffolding or extending was not observed. In half of the classrooms, opportunities
for using concrete objects were not observed. No evidence was found of participants utilizing the
content progressions for extension.
To enact EMTPs, content must be correct. Observations of the use of correct content
knowledge and progressions were varied. Three classrooms utilized correct content for the
duration of the two observations in various ways including anticipating what misconceptions
children might have, what strategies they might use, or what might be challenging as well as

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engaging in conversations during small group work. In an unsuccessful attempt to utilize content
correctly to correct a misconception, Participant 4 explained,
Okay, so what I think I’m hearing you say is that, that when you see a big number is how
you could break it up in a more (inaudible) and the understanding that you’re not
multiplying 3 times 6, but you are saying 3 times 60, right?
Participant 4 then abandoned after a cursory explanation. Other misconceptions were left
uncorrected completely. A further point is that while Participant 1 engaged with the content
correctly, it was more procedural than conceptual. Utilizing the content progressions for
extension was not observed.
Classrooms which enact EMTPs must be free of bias according to race, culture, and other
identifying characteristics. Bias according to race and culture was not evident in any of the six
classrooms. However, bias according to language or learning needs was present in only two of
the classrooms. In Participant 1’s classroom, some children were completely guided through the
process, sometimes with the learning support teacher holding the pencil over the child’s hand.
While this could be considered a physical scaffold, it was noted that the same child was able to
write easily without support. In Participant 3’s classroom, bias according to race, culture or other
characteristics was not present on Day 1 when the children attending math lab were present and
working together in heterogeneous groupings. On Day 2 in Participant 3’s classroom, however,
bias towards children with learning needs was evidenced when the teacher utilized closed
questioning to scaffold their thinking. In four of the six classrooms, no bias according to learning
or language needs was observed. Participant 2 ensured that the ideas of all children were
validated, positioning them as capable and competent, by selecting work from children with
learning needs. This was illustrated by the participant asking the child to share their thinking

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with the class multiple times over two days to validate her thinking and to demonstrate that all
are expected to think deeply and communicate clearly with precise language. In Participant 5’s
classroom, even those children who appeared to be struggling were approached with assessing
and advancing questions by the participant, holding them to high standards. While expectations
were high for all children in several classrooms, asset-based language was only observed in
Participant 1’s classroom when they exclaimed, “Trust yourself! You can do it!”
Summary. While various components of enacting equitable mathematics practices
presented as assets, others presented as needs. In general, it was observed that participants are
engaging in a variety of practices that are equitable, yet generally they cannot articulate why the
practices are equitable. Furthermore, many practices were utilized without achieving the intent.
In all but one classroom, a 3-part framework was used; however, in each of the parts, teachers
were not fully and successfully able to utilize strategies that support equity. For example, when
engaging children in discourse, one goal is to build collective meaning while another goal is for
children to defend arguments, critique reasoning, and build on one another’s thinking. While
engaging children in discourse was attempted, the goal was not fully met. As described
previously, in most cases, exchanges between participant and child dominated rather than
exchanges between children. Hence, mostly the individual who was sharing benefitted. Overall,
there was not a strong sense that content was utilized to anticipate or engage with children to
scaffold or extend. Findings revealed that most tasks were inauthentic, did not build on
children’s experiences, and none helped children understand, critique, or change the world. Joy,
wonder, beauty, and curiosity were largely absent as well. Therefore, while some practices
proved to be assets, overall, this knowledge influence is considered a need.

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Results and Findings for Motivation Causes
Motivation, indicated by active choice, persistence, and effort, is required to accomplish
goals (Rueda, 2011, Schunk et al., 2009). Examining both internal and external impacting
factors, as well as the above-mentioned indicators, helps organizations understand why people
are, or are not, able to accomplish what they aim to do. In order to determine whether teachers
had the motivation to enact EMTPs, two motivational influences were examined for this study.
Influence 1: Teacher Efficacy
Teachers need to believe that they are capable of enacting equitable teaching practices in
their classrooms.
Interview Findings
Overall, most respondents shared a belief that they can enact EMTPs in their classrooms
that allow all students to have access to deep mathematics learning and positively impact math
identity. Two participants stressed the importance of teaching all children in the classroom
together and not having any children pulled from the classroom for either support or extension.
Participant 2 explained that,
I kind of fight to keep kids in my room when I'm teaching all subjects, but math too,
because I feel like when kids are pushed out to you know, learn from someone else, it’s
like you don't belong here you have to go somewhere else and that I feel like that's a
message that I don't think is intended to be sent. But I think it's almost like subconscious.
I know I used to feel that way, like I can't learn with everybody else. So, I have to go
somewhere else.
Alternatively, one teacher who does not feel efficacious in teaching math stated that all needs of
children could only be met with outside support teachers such as math lab and learning support

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and that professional learning on strategies for meeting all children’s needs was needed. While
Participant 2 and 4 described feeling less efficacious in math teaching than other subjects, they
still felt motivated to learn. Participant 4 stated that they still “have a lot to learn,” but feel
supported enough at the school to persist, especially with the support of classroom coaching.
However, Participant 4 stressed the need for a math curricular resource that supported teachers in
meeting the needs of all students to increase teacher efficacy.
Participant 2 felt that while their knowledge of mathematics was less than other subjects,
their philosophy of teaching helped them to feel efficacious in any subject area. Similar to
Participant 4, Participant 2 also stressed that coaching by an expert and collaboration with a
trusted colleague had increased feelings of teacher efficacy while teaching at GVRIS.
Suggestions were made that coaching and trusted collaboration be made more readily available
to support teacher efficacy.
Observation
In five of six classrooms, participants actively chose to, and persisted in, engaging in
practices they believed to be equitable. They continued with this despite acknowledging that
overall, they did not feel the math curricular resources or other support provided by the school
helped them in doing this. Nevertheless, these five teachers demonstrated motivation through
choice by utilizing appropriately challenging tasks. They also demonstrated persistence and
mental effort in attempting to engage students in sense-making and discourse and attempting to
have high expectations for all through scaffolding without lowering the cognitive demand. This
was evident in the questions used to support student thinking.
Summary

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Interview and observational data aligned for this assumed motivational influence. While
not all teachers were successful in enacting equitable mathematics practices as evidenced by
observations, five of six teachers demonstrated high levels of motivation through persistence and
choice. These five teachers also felt confident in their abilities to meet the needs of all students in
their classrooms. Therefore, this assumed motivational influence is considered an asset. While
considered an asset, all teachers made organizational suggestions for increasing teacher efficacy.
These will be shared in the assumed organization influences section.
Influence 2: Collective Efficacy
Teachers need to believe that their collective effort will positively affect student learning
in mathematics.
Interview Findings
Five of six teachers reported that they do not feel confident that the work done as a
professional learning community (PLCs) supports their collective effort in positively affecting
student learning in mathematics. All five of these teachers relayed that the work done in PLCs is
more cooperative than collaborative. They described using PLCs as a time complete unit
overviews rather than look at student work and discuss successful and unsuccessful teaching
strategies. It was reported that teachers are willing to share ideas, but ultimately teachers take on
individual responsibility as each goes back to their classroom and utilizes the resources and
teaching practices of their choice. Participant 2 also questioned if a team of 15 could even
maintain collective responsibility because of the size.
Participant 1 shared a strategy that was utilized several years ago to support collective
ownership of learning, walk-to-math, albeit unsuccessfully. In this model, children were grouped
according to data and shuffled amongst teachers from the onset of a unit. It was reported that this

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model did not only impact student learning, but it also led to tracking practices where children
were labeled and felt ashamed.
Observation
Observations were not conducted for this influence.
Summary
Collective efficacy was found to be a need based on the data gathered during interviews.
Participants described a lack of collective efficacy as they did not feel that the work done in
PLCs contributed successfully to feeling a sense of collective responsibility. Participants
suggested that the school takes the time to focus on mathematics, giving time to PLCs to
collaborate and engage in professional learning to improve their collective efficacy.
Results and Findings for Organization Causes
In order to better understand the ways in which organizations operate and why
organizational goals may not have been met, it is necessary to examine cultural models and
settings (Rueda, 2011). The understanding of how an organization works is referred to as a
cultural model. Cultural settings, on the other hand, are the ways in which organizational policies
are enacted. Three organizational influences were examined to better understand GVRIS’s
cultural models and settings. For this study, evidence for these organizational influences was
examined to better understand how teachers feel the school supports or inhibits their ability to
enact EMTPs.
Influence 1
The organization needs to provide a common vision and philosophy to support an
elementary teacher’s enactment of EMTPs.
Interview Findings

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When asked what GVRIS believed to be important in teaching and learning
mathematics, participants responded in various ways, indicating that a common vision and
philosophy is needed. Half of participants suggested that the standards and benchmarks were
most important. Other responses ranged from, “I don’t know that it’s that easy to identify … .
That can vary depending on who is spoken to” to computation and accuracy being most
important. One participant articulated that teachers “have the capacity to kind of mold their own
lessons around it as long as you are staying true to the benchmarks and the math practices.”
Participant 6 noted that they believed that “it’s important for kids to think deeply about math, to
have rich problems and opportunities to explore, to use concrete materials” but didn’t know how
widespread this belief was. Participant 6 continued, “I know that where we are at the moment,
there’s not truly a math identity for the elementary school.” Participant 5 explained that teaching
math as GVRIS has been frustrating as “the overall core philosophy has not stayed strong or
aligned.”
Observation
While there was a range of practices observed in the six classrooms, in five of six
classrooms participants attempted to engage children in lessons that promoted problem-solving
and sense-making using collaborative structures and discourse. Varied success with
implementation of these lessons was observed, and many facets of equitable math classrooms
were missing, but it was clear that most participants believed that lessons that promote problem-
solving and sense-making are important.
Summary
While it was observed that most participants had similar ideas of what was important in
mathematics classrooms, the enactment of these strategies was varied, and several components of

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equitable mathematics classrooms were missing. Consequently, having a common vision and
philosophy is necessary. Interview data also revealed that having a clear vision and philosophy is
a need. Participants suggested that having a clear goal of what they were trying to accomplish in
the mathematics classrooms would be useful. Recommendations were also made to ensure clarity
of how a math vision connects to other initiatives and disciplines. Further to this, it was
suggested that having clear goals would bring focus to the work, thereby supporting teachers in
enacting equitable opportunities for children in mathematics. Participant 3 summed the need up
nicely by stating,
I think there have to be some common things that you hold your hat on to and say, as a
school, these are the things that we consider to be important. So, really clarifying what is
the key focuses for your school, and these are things we're going to hang our hat on. The
more I think those things are emphasized and driven and the focus of what you do on a
daily basis, the more they become what teachers then want to improve on. Because those
are the things that we say we're going to focus on and we want to do, but when you have
so many different areas and possibilities of what we could do, and what could be
possible, there's no consistency and everybody has an opportunity to take areas that they
find interesting to themselves.
Influence 2
The organization needs to provide ongoing professional learning to support an
elementary teacher’s enactment of EMTPs.
Interview Findings
One common theme that emerged from the interview findings was that participants felt
that the school treats them as professionals and allows for autonomy, which is appreciated.

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Additionally, all participants felt that the school encourages teacher learning. However, 100% of
participants did not feel that there was a culture of learning at the school. To remedy this, it was
suggested that the school needs to build in time, structures, and support to create trusting
relationships for learning. It was also noted that support for teachers in terms of math
professional learning was inequitable and inconsistent.
Further, all six teachers agreed that maintaining a focus in general, and specifically on
math learning, was a priority for GVRIS. They described a school with too many initiatives and
top-down decisions about what to learn and how to use PLC time. Participants argued that
because of this lack of focus, participants have not had opportunities for sustained and ongoing
math professional learning nor have PLCs been used for collaborative math learning to best meet
the needs of their children.
All teachers also felt that while there is support available from a coach when needed, it
needs to be provided proactively rather than sought out. Participant 2 described their experience
as a teacher at GVRIS as “frustrating” but continued that they didn’t “feel that way as much now
… because I sought out the coaching.” Participant 2 further described how coaching supported a
feeling of efficacy in math teaching by noting, “That one experience shifted my math experience
or my teaching math experience.” Participant 2 also stressed, however, that had they not felt
comfortable or been proactive in seeking out coaching, they would not have received it.
Additionally, Participant 6 stated that being on the math distributed leadership team supported
their math teaching as they had access to additional math professional learning opportunities, but
that this caused inequities for others as they didn’t receive the same professional learning.
Observation
Observations were not conducted for this influence.

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Summary
Interview findings clearly suggest a lack of ongoing professional learning to support
teachers in enacting EMTPs. The absence of focus as well as proactive and equitable support was
reported. Consequently, this influence is considered a need. The school would benefit from
finding time to focus on math learning based on a common vision, but with teacher choice.
Influence 3
The organization needs to provide standards-based curriculum instructional materials
with support for implementation to support an elementary teacher’s enactment of EMTPs.
Interview Findings
All six teachers agreed that there was not a common resource and teachers have the
autonomy to select resources if standards are adhered to. Participant 5 described this as having to
“forge my own teaching plan.” Participant 1 shared that the school is well resourced and that
being able to be flexible is appreciated. The other five participants also felt appreciation for
having some autonomy to meet their student’s needs and stated that this was a necessity.
Participant 4 felt strongly that a program was required for successful enactment of EMTPs,
stating that it was difficult to know how to support a range of learners' needs without a resource
aligned to the standards and school-wide philosophy. While all six participants suggested that
consistency and a common vision is needed, only half of the participants revealed that having a
common resource would be useful to their math teaching.
Observation
Observations found that all participants were utilizing different resources even within the
same grade level, which resulted in different types of learning experiences for children. One
participant used technology such as Freckle for conceptual learning, videos and songs found on

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the internet to teach, IXL for practice, and quizzes converted into google forms for immediate
feedback on the correct answer. Meanwhile, in a classroom of the same grade level, a participant
engaged children in a problem-based lesson where children were using their own strategies to
solve problems.
In another classroom, a participant was using what has been considered a reputable
resource by the school, but the teacher did not clearly understand the content being taught and
thus misunderstood the intent of the lesson. In another classroom, a participant was utilizing a
new resource the school is considering for adoption. That participant utilized the lesson plan, but
there was a clear indication that they were unsure when to allow children to inquire versus when
they should move the children towards a synthesis or clarify the misconceptions. This was
previously noted in the content section. As mentioned previously, lessons were implemented
with various success in terms of EMTPs. This indicates that a resource, while not suggested by
teachers, is necessary to maintain equity of opportunity.
Summary
Findings from interviews and observational data revealed the need for a standards-based
curriculum instructional materials. A lack of consistency in beliefs and practices, as well as
misconceptions in both content and purpose of learning goals, was noted with some participants.
In addition to having a common resource, participants need support and collaborative
conversations to be clear on the mathematics they are teaching and how to use the resources with
children. Consequently, the findings support the assertion that this influence is considered a
need.
Summary of Validated Influences
Knowledge

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Table 8 shows the knowledge influences for this study and their determination as an asset
or a need.

Table 8
Knowledge Assets or Needs as Determined by the Data
Assumed knowledge influence Asset or need
Factual and conceptual
Teachers know and understand goals for
mathematics and math teaching.
Need
Teachers know and understand what equitable
math teaching practices are and how they
promote equitable opportunities for learning.
Need
Teachers will know and understand
mathematical content knowledge and
progressions.
Need
Procedural
Teachers know how to enact equitable math
teaching practices in their classrooms.
Need

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Motivation
Table 9 shows the motivation influences for this study and their determination as an asset
or a need.

Table 9
Motivation Assets or Needs as Determined by the Data
Assumed motivation influence Asset or need
Teacher efficacy
Teachers need to believe that they are capable of
enacting equitable teaching practices in their
classrooms.
Asset
Collective efficacy
Teachers need to believe that their collective
effort will positively affect student learning in
mathematics.
Need

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Organization
Table 10 shows the organization influences for this study and their determination as an
asset or a need.

Table 10
Organization Assets or Needs as Determined by the Data
Assumed organizational influence Asset or need
The organization needs to provide a common
vision and philosophy to support an
elementary teacher’s enactment of EMTPs
Need
The organization needs to provide ongoing
professional learning to support an
elementary teacher’s enactment of EMTPs.
Need
The organization needs to provide standards-
based curriculum instructional materials to
support an elementary teacher’s enactment of
EMTPs.
Need

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The results in this chapter represent the knowledge and motivation of teachers attempting
to enact EMTPs and their perceptions of how the organization supports or hinders their ability to
do so. Chapter 5 will present an integrated plan with recommendations for solutions for these
influences based on research, an implementation plan, and ways to measure success.

108
Chapter Five: Recommendations and Evaluation
Mathematics plays a key role in being prepared for the 21st century and ensures
opportunities for success in life (English & Gainsburg, 2015). Consequently, having equitable
access to deep mathematical learning for all children is a necessity. Unfortunately, there are
many reasons why some children are not afforded these opportunities. During the most recent
math review at Green View River International School (GVRIS), it was found that inconsistent
enactment of equitable mathematics teaching practices (EMTPs) resulted in inequitable
opportunities for all students to engage in deep math learning. As a result, this study aimed to
determine the gaps in teacher knowledge, motivation, and organizational influences that ensure
equitable access to mathematics for all. An adapted gap analysis was employed as a framework
(Clark & Estes, 2008). The questions that guided this study were as follows:
1. What is the current status of teachers’ knowledge and motivation related to enacting
equitable mathematics teaching practices, or those teaching practices that provide
access to deep mathematics learning for all learners?
2. How do organizational factors influence teachers’ capacity to enact equitable
mathematics teaching practices?
Participants in this study included six teachers from grades 3–5 as was set out in the
criteria. Each participant engaged in two semi-structured interviews, revealing perceptions of
knowledge, motivation, and organizational influences. Two classroom observations per
participant were also conducted to triangulate data.
The preceding chapter, Chapter 4, provided a synthesis of findings related to the
knowledge, motivation, and organizational influences which support equitable opportunities for
math learning. The purpose of Chapter 5 is twofold. It will offer evidenced-based solutions and

109
recommendations for knowledge and motivation gaps as well as organizational barriers. This
will be followed by an integrated implementation and evaluation plan utilizing the new world
Kirkpatrick model to ensure success of proposed outcomes (Kirkpatrick & Kirkpatrick, 2016).
Recommendations to Address Knowledge, Motivation, and Organization Influences
The following sections detail and prioritize the needs of the knowledge, motivation, and
organization influences based on the findings in Chapter 4. Each high-priority need is aligned to
principles which inform context-specific recommendations.
Knowledge Recommendations
According to Anderson and Krathwohl (2001), four types of knowledge exist. These are
factual, conceptual, procedural, and metacognitive. This study examined factual, conceptual, and
procedural knowledge influences related to the successful enactment of EMTPs. Factual and
conceptual knowledge influences included:
• knowing and understanding the goals for mathematics and mathematics teaching; and
• knowing and understanding what equitable math teaching practices are and how they
promote equitable opportunities for learning and knowing and understanding
mathematical content knowledge and progressions.
One procedural influence, knowing how to enact equitable math teaching practices, was
also examined. A synthesis of findings based on interview and observational data revealed all
three factual and conceptual knowledge influences as validated as gaps. The procedural
knowledge influence in this study was also determined to be a need.
While evidence was presented that teachers know and understand one goal of
mathematics, positioning children as knowers and doers of mathematics, varying levels of
understanding were observed. Other goals of mathematics were not evident in the findings,

110
namely understanding the beauty and wonder of mathematics and understanding, critiquing, or
changing the world. Data also revealed that while teachers were able to describe some equitable
mathematics practices, most were not able to articulate what made them equitable. Findings
related to content knowledge and progressions were varied. Most teachers could correctly
articulate content knowledge in the interview; however, some of these teachers engaged
incorrectly with content during the observations. Furthermore, the use of learning progressions to
support student learning was not observed in most classrooms. Finally, while a strong attempt
was made, some identified EMTPs were enacted with varying levels of success and others were
not present during the observations.
As a result of these findings, all four knowledge factors have been validated as gaps,
therefore hindering the enactment of EMTPs that support deep math learning for all. Table 11
lists the causes, level of priority, principles, and recommendations for the knowledge gaps.
Following the table, a detailed discussion for each high priority cause and recommendation and
the literature supporting the recommendation is provided.

Table 11
Summary of Knowledge Influences and Recommendations
Assumed knowledge
influence
Asset or
need
Priority
high or
low
Principle and
citation
Context-specific
recommendation
Factual and conceptual
Teachers know and
understand the goals
for mathematics and
mathematics
teaching.
Need High How individuals
organize
knowledge
influences how
they learn and
apply what they
know (Schraw &
McCrudden,
2006).
Provide readings on the
goals for
mathematics
education. Engage in
team discussions to
identify and
understand important
points (Shraw &
McCrudden, 2006).

111
Assumed knowledge
influence
Asset or
need
Priority
high or
low
Principle and
citation
Context-specific
recommendation
Increasing germane
cognitive load by
engaging the
learner in
meaningful
learning and
schema
construction
facilitates
effective learning
(Kirschner et al.,
2006).
Have learners
outline, summarize,
or elaborate on the
material (Mayer,
2011)

Teachers know and
understand what
equitable math
teaching practices are
and how they
promote equitable
opportunities for
learning.
Need High Active learning
engagements
rather than
passive processing
encourages
learning (Rueda,
2011).

Increasing germane
cognitive load by
engaging the
learner in
meaningful
learning and
schema
construction
facilitates
effective learning
(Kirschner et al.,
2006)

Decreasing
extraneous
cognitive load by
effective
instruction
Engage teachers in
lessons as students
where they can
personally engage in
the EMTPs and then
discuss how they
support equitable
opportunities for
learning
mathematics.
Explicitly share what
equitable
mathematics
practices are and
how they promote
equitable
opportunities for
learning by
providing concrete
examples (Anguinis
& Kraiger, 2009).
Connect to prior
knowledge (Mayer,
2011), the purposes
and goals of
mathematics.
Provide a graphic
organizer with
visuals that lists the
practices and how
they promote

112
Assumed knowledge
influence
Asset or
need
Priority
high or
low
Principle and
citation
Context-specific
recommendation
(particularly when
intrinsic load is
high) enables
more effective
learning
(Kirschner et al.,
2006)
Schraw and
McCrudden
(2006) suggest
that how
individuals
organize
knowledge
influences how
they learn and
what they apply.
equitable
opportunities for
learning.

Teachers will know and
understand
mathematical content
knowledge and
progressions.
Need Low Active learning
engagements
rather than
passive processing
encourages
learning (Rueda,
2011)
Collective efficacy
could likely be
improved by
teachers working
collaboratively to
learn, problem-
solve, and
successfully
implement change
together (Goddard
et al., 2000).
When planning units,
discuss required
grade level content
as well as the content
that comes before
and after.

Engage in lessons as a
team to
collaboratively plan
explicit scaffolds and
extensions to support
learners at all levels
and discuss common
misconceptions and
challenges and plan
explicit strategies,
including questions,
to support student
understanding.
Procedural
Teachers know how to
enact equitable math
teaching practices in
their classrooms.
Need High To develop mastery,
individuals must
acquire
component skills,
practice
Provide a series of
professional learning
sessions over time to
introduce ways to
enact EMTPs.

113
Assumed knowledge
influence
Asset or
need
Priority
high or
low
Principle and
citation
Context-specific
recommendation
integrating them,
and know when to
apply what they
have learned
(Schraw &
McCrudden,
2006).
Learning and
motivation are
enhanced when
learners set goals,
monitor their
performance and
evaluate their
progress towards
achieving their
goals. (Ambrose
et al., 2010;
Mayer, 2011)
Provide immediate
feedback and
reinforcement
(Tuckman, 2009).
Feedback that is
private, specific,
and timely
enhances
performance
(Shute, 2008).
Modeling to-be-
learned strategies
or behaviors
improve self-
efficacy, learning,
and performance.
Learners acquire
new behaviors
through
demonstration and
modeling (Denler,
et al., 2009).
Modeled behavior is
more likely to be
Couple with this will
a look-for document
that will be used for
monitoring progress,
so teachers know
specifically what is
expected.
After each session,
teachers will set
goals according to
new learning. They
will monitor their
progress using the
look-for tool.

Provide
feedback/coaching
shortly after training
on specific behaviors
related to enacting
EMTPs.

Engage teachers in
collaborative lab
sites where
behaviors and
strategies to enact
EMTPs will be
modeled or
demonstrated.
Engage teachers in
lessons as students
where they can
personally engage in
the EMTPs and then
discuss how they

114
Assumed knowledge
influence
Asset or
need
Priority
high or
low
Principle and
citation
Context-specific
recommendation
adopted if the
model is credible,
similar, and the
behavior has
functional value
(Denler et al.,
2009).
Collective efficacy
can likely be
improved by
teachers working
collaboratively to
learn, problem-
solve, and
successfully
implement change
together (Goddard
et al., 2000).
support equitable
opportunities for
learning
mathematics.

Knowledge Solutions
Findings revealed that all four knowledge influences were validated as gaps. Three of
these influences are considered of high priority and will be detailed in the following sections,
which are divided by influence. Evidence-based solutions are provided for each influence to
support teachers in learning the required declarative and procedural knowledge required for the
enactment of EMTPs.
Teachers know and understand the goals for mathematics and mathematics teaching
(declarative). To ensure all children are afforded opportunities for deep math learning, teachers
must know and understand the goals of mathematics and mathematics teaching. Mathematics
should be viewed as a flexible discipline that allows children to explore and make sense in a
variety of ways, leading to children envisioning themselves as knowers and doers of mathematics

115
(NCTM, 2020). Furthermore, the wonder and beauty of math should be revealed, allowing
children to enjoy and appreciate mathematics and visualize themselves in it (Boaler, 2008;
California Department of Education et al., 2021; NCTM, 2020). Math should not be seen as
something done at school. Instead, using mathematics as a tool to understand, critique, and
change the world is an essential purpose of mathematics teaching and will lead children to not
only view themselves as knowers and doers of mathematics, but it will also prepare them to
better understand society and participate in a democratic society (Yolcu, 2019). Only when
teachers know and understand these important goals can they afford all children opportunities to
engage in deep math learning and see themselves as mathematicians. Therefore, priority has been
assigned to closing this influence gap.
Schraw and McCrudden (2006) suggested that the way individuals organize knowledge
influences how they learn and what they apply. Providing organizers, then, would support
learning. Additionally, engaging individuals in meaningful learning and schema construction
facilitates learning (Kirschner et al., 2006) as does teacher collaboration (Gasser, 2011; Harris et
al., 2017; Kleickmann et al., 2013; Lin et al., 2013; Loughland & Nguyen, 2020; Louie, 2020)
supporting the idea of the engagement of teachers in team discussions. To facilitate the
knowledge of goals of mathematics and math teaching along with understanding of how they
present in elementary classrooms, several context-specific solutions are provided.
To begin, it is recommended to provide teachers with readings on the goals of
mathematics alongside a graphic organizer to outline and summarize the important points
(Mayer, 2011). Mayer (2011) also suggested that learners elaborate on material. Therefore, it is
recommended that following individual reading accompanied by outlining and summarizing,
teams should engage in reflection and discussion about the important points, elaborating on the

116
ideas by discussing scenarios where they have observed these purposes and how they support
equitable mathematics learning. Visuals accompanying each goal would also support learning.
Teachers know and understand what equitable math teaching practices are and how they
promote equitable opportunities for learning (declarative). If teachers do not know and
understand what equitable math teaching practices are, they cannot enact them. Therefore, this
influence is considered of high priority.
Several teaching practices have been found to support equitable opportunities for all
students to develop deep math understanding and potentially develop positive math identities as
a result. An adapted framework of equitable teaching practices has been created for this study
building on the work of Huinker and Bill (2017) in their mathematics teaching framework and
Bartell et al. (2017)’s equitable mathematics teaching practices. These include the following:
• implementing tasks that promote reasoning and problem-solving
• facilitating meaningful mathematical discourse
• drawing on students’ funds of knowledge
• establishing norms for participation
• positioning students as capable
• attending explicitly to race, culture, and other characteristics
• attending to students’ mathematical thinking
In classrooms where said practices are in place, children can find joy in mathematics and
learn deeply because they see themselves in the mathematics and realize that the authority rests
within themselves as meaning makers. Together with the entire classroom community, they
decide what counts as appropriate solutions and explanations (Cobb & Hodge, 2002; Gutierrez et

117
al., 1995). Consequently, they see themselves as capable of making meaningful contributions to
both the classroom and the world.
As stated previously, Schraw and McCrudden (2006) suggested that how individuals
organize knowledge influences how they learn and what they apply. Additionally, Kirschner et
al., (2006) indicated that decreasing extraneous cognitive load by effective instruction enables
more effective learning. A recommendation then is to provide visual aids that explicitly outline
equitable mathematics practices and how they promote equitable opportunities for learning with
concrete examples (Anguinis & Kraiger, 2009). Rueda (2011) proposed utilizing active learning
engagements rather than passive processes for encouraging learning. Based on this principle, a
recommendation is to engage teachers in lessons as students where they can personally
participate in the EMTPs and then discuss how the EMTPs support equitable opportunities for
learning mathematics. Finally, increasing germane load by engaging the learner in meaningful
learning facilitates effective learning (Kirschner et al., 2006). One way to engage individuals in
meaningful learning is to make the learning personal, or present learning in the context of a
familiar situation (Mayer, 2011). A recommendation for supporting teachers to better understand
EMTP through making personal connections is to facilitate a restorative justice circle in which
teachers share their personal math stories, and the feelings that accompany their stories, and
connect them to equitable and inequitable mathematics teaching practices.
Teachers know how to enact equitable math teaching practices in their classrooms
(procedural). Knowing what EMTPs are is important, but without knowing how to enact them,
one cannot support deep math learning for all. Enacting EMTPs is complex. There are many
strategies which support children learning mathematics through doing mathematics so that they
can see themselves as capable. One strategy, utilizing a 3-part problem-based lesson structure to

118
frame the collaborative inquiry, where children are positioned as capable, is necessary (Van de
Walle, 2012). Within the framework, children are launched into the learning task with
excitement and curiosity, followed by the engage portion where children collaborate and
communicate to make sense of the mathematics and contribute meaningfully, and concluding
with the summary portion where the teacher utilizes the children’s’ thinking to facilitate
discourse to create collective meaning.
As this process is complex, it is important to consider Schraw and McCudden’s (2006)
idea that to develop mastery, individuals must acquire component skills, practice integrating
them, and know when to apply what they have learned. To ensure successful learning, a
recommendation is to provide a series of professional learning sessions over time to introduce
ways to enact EMTPs. It is recommended that these professional learning sessions be
accompanied by a clearly articulated look-for tool so that teachers are clear about what they are
aiming for, can set goals, and monitor their growth (Ambrose et al., 2010; Meyer, 2011).
In addition to breaking down the skills in a series of professional learning sessions, a
recommendation is to engage teachers in collaborative lab sites, where to-be-learned strategies
are modeled. Denler et al. (2009) suggested that demonstration and modeling help learners
acquire new behaviors. Furthermore, modeling new behaviors is said to improve self-efficacy,
learning, and performance (Denler, et al., 2009). Also, important to note is that modeled behavior
is more likely to be adopted if the model is credible and similar, and the behavior has functional
value (Denler et al., 2009). As such, lab sites should be led by PLC coaches who teach in the
same grade levels as the observers, when possible.
According to Tuckman (2009), feedback and reinforcement enhance learning. As such, it
is recommended that coaching and feedback be provided immediately after training on specific

119
behaviors related to enacting EMTPs as indicated on the look-for tool. Shute (2008) also
suggests that this feedback is private, specific, and timely.
Engaging teachers in collaborative engagements for learning is a final recommendation
for this influence. Collective efficacy can likely be improved by teachers working collaboratively
to learn, problem-solve, and successfully implement change together (Goddard et al., 2000). In
addition to collaborative professional learning sessions, a recommendation is to engage teachers
collaboratively in lessons as students where they can personally participate in the EMTPs and
then discuss how the teaching moves support equitable opportunities for learning mathematics.
Facilitating detailed, collaborative lesson planning and reflective conversations is also suggested
to enhance learning (Loughland & Nguyen, 2020) and is part of the recommendation.
Motivation Recommendations
Motivation is required to accomplish goals (Rueda, 2011). Three indicators of motivation
have been identified: active choice, persistence, and effort. These indicators must be considered
along with external and internal factors that impact motivation, to support the attainment of
goals. Self-efficacy, the belief that one is capable of performing a task, impacts motivation
(Schunk, 2020). Schunk and DiBenedetto further elaborated on self-efficacy and named teacher
efficacy as a construct impacting the success of motivation and performing in teaching. More
specifically, this construct refers to the teacher’s belief that they can help students learn (Schunk
& DiBenedetto, 2016). Teacher efficacy and collective efficacy were examined in relation to
being able to successfully enact EMTPs.
Teacher efficacy was measured by interviews and observations and were found to be high
among most participants in both data sources. This was measured by the active choice to attempt

120
to engage in EMTPs and persistence and effort in doing so. As such, teacher efficacy was not
found to be a need within this study.
Collectively efficacy, however, was determined to be a need as indicated by most
teachers through interviews. While each individual teacher felt they could impact the learning of
their individual students, they did not feel that collectively the focus or goal of their Professional
Learning Community (PLC) was to collectively meet students’ needs. Table 12 lists the
motivation causes, priority, principles, and recommendations for supporting the development of
collective efficacy. Following the table, a detailed discussion with a research-based
recommendation is provided.

Table 12
Summary of Motivation Influences and Recommendations
Assumed motivation
influence
Asset or
need
Priority
high or
low
Principle and
citation
Context-specific
recommendation
Teacher efficacy
Teachers need to
believe that they are
capable of enacting
equitable teaching
practices in their
classrooms.
Asset NA High self-efficacy
can positively
influence
motivation.

Collective efficacy
Teachers need to
believe that their
collective effort will
positively affect
student learning in
mathematics.
Need Low Collective efficacy
can likely be
improved by
teachers working
collaboratively to
learn, problem-
solve, and
successfully
implement change
together (Goddard
et al., 2000).
Provide structures and
time for PLCs to
plan, implement, and
reflect on EMTPs
using student
learning data.
Create a community of
learners where
everyone supports
everyone else’s

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Assumed motivation
influence
Asset or
need
Priority
high or
low
Principle and
citation
Context-specific
recommendation
Attempts to learn
(Yough &
Anderman, 2006).

Collective Efficacy Solutions
Goddard et al., (2000) suggested that collective efficacy could likely be improved by
teachers working collaboratively to learn, problem-solve, and successfully implement change
together. As participants in this study shared that the purpose or goal of their PLC was other than
collective responsibility for children to learn deep mathematics, one recommendation is to
provide structures and focused time for PLCs to plan, implement, and reflect on EMTPs using
student learning data. This would allow for the creation of a community of learners where
everyone supports everyone else’s attempts to learn (Yough & Anderman, 2006).
Organization Recommendations
Organizations are made up of cultural settings and models. Examining these constructs
helps organizations to better understand how they operate and why goals may not have been met.
For this study, three cultural settings were examined that support the enactment of EMTPs.
These included the following in support of teachers to enact EMTPs:
• providing a common vision and philosophy
• providing ongoing professional learning
• providing standards-based curriculum instructional materials with support

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All three influences were validated as gaps; therefore, the lack of these influences can be
said to be hindering teachers’ ability to ensure all students engage in deep math learning. As
such, each influence was determined to be of high priority as they are all interrelated.
Observations illustrated the possibility of a shared vision between five of six participants
in that their teaching practices supported deep learning for all with varying levels of success.
However, remarks during the interviews revealed that the organization does not have a common
vision. It was proposed by participants that having a common vision and philosophy would bring
focus to the school. Many felt overwhelmed not only by the lack of clarity in mathematics, but
the many priorities in this school. By having a common vision and philosophy, it was suggested
that individuals would feel more successful in their mathematics teaching and PLCs could find
focus, thereby increasing collective efficacy.
The need for professional learning was also validated as an organizational barrier. While
teachers felt the school encouraged learning and indicated that support was available through
coaching and professional learning, the sporadic nature, availability by request only, and
inequitable opportunities for professional learning did not successfully support all teachers in
their ability to effectively teach mathematics.
While only one teacher suggested that having a common standards-based resource was
necessary for their success, it was still validated as a gap because interview and observational
evidence clearly suggested that there was not a common resource to guide teachers, as deemed
important by the literature. Some teachers reportedly appreciated the autonomy to choose their
own resources based on the needs of their students. The fear was mentioned that by having a
common resource, all teachers will have to teach the same way and would not be able to respond
to the needs of their children. This concern aligns closely with the idea that schools must not

123
implement resources with fidelity, but with integrity, which was found to be important by
(Huinker & Bill, 2017). To elaborate, teachers must be able to use their professional judgment
and knowledge of students to guide the resource implementation rather than follow the resource
blindly. While maintaining some autonomy to meet children’s needs is necessary, the various
levels of quality materials utilized in different classrooms is potentially creating equity gaps for
students. Therefore, a resource which can be implemented with a clear philosophy and vision, is
needed. Additionally, it was observed that while some teachers were using resources considered
of high quality, their implementation was unclear in some cases, indicating that teacher
collaboration and focused support around a common resource is necessary (Charalambous &
Hill, 2012) for implementing with integrity.
As a result of these findings, all three organizational influences have been found to hinder
the ability of teachers to enact EMTPs and ensure deep math learning for all. Table 13 lists the
organizational causes, priority, principle and recommendations. Following the table, a detailed
discussion for each high priority cause, a recommendation, and the supporting literature is
provided.

Table 13
Summary of Organization Influences and Recommendations
Assumed organization
influence
Priority
high or
low
Principle and
citation
Context-specific
recommendation
The organization needs to
provide a common vision
and philosophy to support
an elementary teacher’s
enactment of EMTPs
High Effective change
efforts ensure that
all key
stakeholders’
perspectives
inform the design
and decision-
Ensure the team
includes a variety of
people with diverse
thinking. Share and
get feedback
anonymously.

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Assumed organization
influence
Priority
high or
low
Principle and
citation
Context-specific
recommendation
making process
leading to the
change.
A common vision is
essential to direct
change (Kotter,
2011).

Effective change
begins by
addressing
motivation
influencers; it
ensures the group
knows why it
needs to change.
(Clark and Estes,
2008).

Develop a common
vision and
philosophy with
clear indicators of
what equitable
teaching and
learning of
mathematics sounds
and looks like
(NCSM, 2019)
Use data from math
review to ensure
there is an
understanding of
why we need a new
vision and
philosophy in
mathematics.
The organization needs to
provide ongoing
professional learning to
support an elementary
teacher’s enactment of
EMTPs.
High Intensive and
ongoing
professional
learning
connected to
practices had a
greater chance of
influencing
teaching practices
than less-
intensive, one-off
professional
development
sessions that
focused on theory.
(Darling-
Hammond et al.,
2009; Garet et al.
2001)

Design professional
development
experiences that are
coherent, sustained,
collaborative,
connect theory and
practice, reflection
on student learning,
planning and
instruction (Darling-
Hammond et al.,
2009).

125
Assumed organization
influence
Priority
high or
low
Principle and
citation
Context-specific
recommendation
Teacher
collaboration is
important for
learning (Gasser,
2011; Harris et al.,
2017; Kleickmann
et al., 2013; Lin et
al., 2013;
Loughland &
Nguyen, 2020;
Louie, 2020).
Support teachers to
ground collaborative
discussions in ideas
about equity by
examining
positionality and
beliefs to think about
who is included or
excluded in the
classroom (Louie,
2020).
Link back to vision and
why change is
needed.
The organization needs to
provide standards-based
curriculum instructional
materials to support an
elementary teacher’s
enactment of EMTPs.
High Effective change
efforts ensure that
everyone has the
resources needed
to do their job,
and that if there
are resource
shortages, then
resources are
aligned with
organizational
priorities (Clark
and Estes, 2008)
Select standards-based
curriculum materials
that align with the
vision and
philosophy (Hill et
al., 2008; NCTM,
2020).

Cultural Settings Solutions
Findings revealed that all three organizational influences were validated. All three are
considered a high priority and will be detailed in the next sections. Evidence-based solutions
based on organizational gaps are provided to support teachers in being able to successfully enact
EMTPs.

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The organization needs to provide a common vision and philosophy to support an
elementary teacher’s enactment of EMTPs. Kotter (2011) argued that a common vision is
essential to direct change. The vision should provide clarity for teachers and leaders to better
understand what equitable mathematics means. Creating a vision with clear indicators of success
is recommended to ensure an equitable mathematics program and to guide strategic planning,
professional learning, and goal setting.
Effective change efforts also ensure that all key stakeholders’ perspectives inform the
design and decision-making process leading to the change. As such, a recommendation is to
gather a committee with a variety of perspectives to engage in this important work. Furthermore,
the team needs to understand why they need to change to be effective (Clark and Estes, 2008).
Therefore, as the team begins, it will be necessary to utilize data collected during the review to
illustrate the need for change.
The organization needs to provide ongoing professional learning to support an
elementary teacher’s enactment of EMTPs. Research suggested several indicators of successful
professional learning programs. Garet et al. (2001) and Darling-Hammond et al. (2009) found
that intensive and ongoing professional learning connected to practice had a greater chance of
influencing teaching practices than less-intensive, one-off professional development sessions that
focused on theory. Professional learning experiences that are coherent, sustained, collaborative,
reflect on student learning, and support planning and instruction were also found to contribute to
successful learning (Darling-Hammond et al., 2009). As a result, it is recommended that the
organization develops a coherent and sustained professional learning plan based on the vision.
Within this plan, professional development sessions need to be collaborative and allow teachers
to build understanding with opportunities to visualize how this learning can be applied in the

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classroom. It has also been noted that providing immediate feedback and reinforcement promotes
learning (Tuckman, 2009). Professional learning sessions, then, should be followed by coaching
and feedback to help teachers make connections between theory and practice and reinforce the
learning from professional development sessions. It is also recommended that PLCS are
provided time and support to collaboratively plan for instruction and reflect on student learning.
Collaborative discussions should be grounded in ideas about equity by examining positionality
and beliefs to think about who is included or excluded in the classroom (Louie, 2020).
The organization needs to provide standards-based curriculum instructional materials to
support an elementary teacher’s enactment of EMTPs. Effective change efforts ensure that
everyone has the resources needed to do their job (Clark and Estes, 2008). Additionally, teachers
must be provided with supportive physical resources for successful enactment of EMTPs (Hill et
al., 2008). Supporting these ideas is the recommendation to select high-quality curricular
materials that align with the standards and vision.
Research also suggests that while quality materials are necessary, support to utilize them
in a way that allows deep mathematical learning for all children is necessary. Teachers cannot
just follow the resource with fidelity, meaning follow it page-by-page. Instead, they must follow
with integrity by ensuring alignment to the vision and through responding to the needs of their
children. Support for utilizing materials is recommended to be built into the cohesive
professional learning plan, including focused time for planning and reflecting in PLCs.
Summary of Knowledge, Motivation, and Organization Recommendations
In order to support teachers to successfully enact EMTPs, three recommendations have
been made to close gaps in knowledge and motivation and three recommendations for
organizational influences were detailed.

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To begin, graphic organizers should be used to outline and summarize relevant text to
facilitate learning of the goals of mathematics and math teaching. Teams should collaboratively
discuss the most important points and engage in reflection to make connections to their practices
in terms of when they have observed these goals in action and how they support equity.
Teachers also need to know what EMTPs are to be successful in ensuring all children
engage in deep mathematical learning. To understand these practices, it is recommended that
visual aids which explicitly outline equitable mathematics practices accompanied by concrete
examples detailing how they promote equitable opportunities for learning are provided. Further,
teachers should engage in lessons as students so they can be immersed in the EMTPs. This
should be followed by a discussion on how the EMTPs support equitable opportunities for
learning mathematics. Finally, to make these practices personal, it is recommended that teachers
connect to their lives through reflection on their mathematical experiences growing up. A
restorative justice circle would provide the opportunity for teachers to share their experiences
with equitable and inequitable mathematics teaching practices along with the impact it had on
them as learners and teachers.
Successfully enacting the EMTPs is also essential. To close this identified procedural
knowledge gap, there are several closely related recommendations incorporating the ideas of
collaboration and feedback. To begin, an introduction of strategies for enacting EMTPs through
a series of coherent collaborative professional learning sessions with coaching and feedback
immediately following is recommended. A look-for tool to guide conversations will allow focus
for goal setting, monitoring of goals, and feedback. Collaborative lab sites where newly
introduced strategies are modeled is an additional strategy that has been recommended to support
teachers in enacting EMTPs. The use of grade level peers to model these practices should be

129
utilized when available. Two further recommendations are to engage teachers collaboratively in
lessons as students where they can personally engage in the EMTPs and then discuss how the
teaching moves support equity and to provide opportunities for teachers to engage in facilitated
collaborative lesson planning and reflective conversations. Recommendations for closing the
collective efficacy gap include providing structures and focused time for PLCs to plan,
implement, and reflect on EMTPs based on student learning data.
All organization influences were determined to be barriers for teachers to enact EMTPs
to support deep mathematical learning for all. As such, several recommendations have been
made for the organization to provide the support needed. To begin, a vision and philosophy with
clear indicators of success that are developed based on input of many stakeholders should be
created. As such, a team should be convened to engage collaboratively in this work.
The development of a coherent and sustained professional learning plan based on the
vision is also recommended. These sessions should be collaborative in nature and connect theory
to practice. Additionally, coaching and feedback should follow the professional development
sessions to support teachers in making connections between theory and practice and to reinforce
the learning from professional development sessions. Finally, PLCs need to be provided time and
support to collaboratively plan for instruction and reflect on student learning. Support includes
clear structures, outcomes, expectations, and roles for focused and purposeful work. To move
beyond coordination to collaboration, training on strategies for collaboration, navigating team
dynamics, and developing a team vision for collaboration should be provided.
A final recommendation for the organization is to select high-quality materials that align
with the standards and the organization’s vision. This should be augmented with support to
utilize the curriculum with integrity rather than fidelity to ensure student needs are met. This

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recommendation should be built into the above-mentioned professional learning plan as teachers
can learn to enact EMTPS through using a quality resource. Reflection on student learning
impacted by the use of the materials and specific teaching practices should be a part of the
professional learning plan as well.
The implementation of these recommendations is complex, so a clear path is necessary.
The leading recommendation is to develop a common vision and philosophy with a look-for
document. The look-for tool will include indicators for what the vision and philosophy look like
in action and will be calibrated between all stakeholder to ensure consistent expectations. All
other recommendations will build from the common vision and philosophy. Following the
development of the vision and philosophy, an aligned curricular resource will be selected to
support consistency. It is primarily through the implementation of this resource that teachers will
learn about the goals of mathematics and math teaching, what the EMTPs are and how they
promote equity, and strategies for enacting the EMTPs. Complimentary individual and
collaborative professional learning will take place as necessary to connect the philosophical and
practical.
Integrated Implementation and Evaluation Plan
Organizational Purpose, Need, and Expectations
GVRIS is committed to providing each student with an American education with a global
perspective as well as developing exceptional citizens prepared for a changing world. To be
prepared for the 21st century, one must maintain deep mathematical knowledge to apply skills to
solve unknown problems (English & Gainsburg, 2015; Gasser, 2011). Additionally, in order to
feel successful in mathematics, one must maintain a positive mathematics identity (Aguirre et al.,
2013). During the most recent math review, it was revealed that children at GVRIS are afforded

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different types of opportunities in mathematics, leading to gaps in deep mathematics knowledge
and possibly resulting in discrepancies in how children view themselves as mathematicians. As a
result, the organizational goal is that by May 2027, all students will demonstrate deep
mathematics learning of content and mathematical practices standards, as well as the learning
aspirations, as measured by unit assessments. To support this student goal, by May 2023,
elementary teachers at GVRIS will consistently enact EMTPs as defined by a school-designed
rubric. Enactment will be measured by formal and informal observations in collaboration with
the instructional coach, administrators, and math leaders. With the consistent enactment of
EMTPs, children will feel empowered as knowers and doers of mathematics and be prepared to
utilize mathematics as they participate in a world that is rapidly changing.
Implementation and Evaluation Framework
Training programs must transfer learning to behavior to be considered effective
(Kirkpatrick & Kirkpatrick, 2016). The new world Kirkpatrick model has been utilized to guide
the development of the implementation and evaluation plan to determine whether the behaviors
resulting from the recommendations have been implemented. This new model has been adapted
from the original 4-level model of evaluation Kirkpatrick model (Kirkpatrick & Kirkpatrick,
2006) and utilizes a backwards-by-design approach, starting with Level 4 to ensure
accountability for results. The new world model offers four levels to evaluate a program: results,
behavior, learning, and reaction.
These four levels together provide a framework for evaluating an effective training
program, determining both the quality and value. Level 4 suggests that the organization has
accomplished what it exists to do according to the mission and vision. Level 3 defines the critical
behaviors of the stakeholders that will significantly impact the desired behaviors, followed by

132
required drivers, or processes and systems that support the implementation of critical behaviors
on the job. On-the-job learning is a part of Level 3 as well, and helps employers and employees
share responsibility for enacting the critical behaviors. Level 2 incorporates the dimensions of
knowledge and skill, attitude, confidence, and commitment to determine if the gap between
learning and performance has been closed. Finally, Level 1 measures how the participants react
to the learning event. Level 1 includes customer satisfaction, relevance, and engagement. The
remaining sections detail each of the four levels of the new world Kirkpatrick model alongside
evaluation measures for each recommendation.
Level 4: Results and Leading Indicators
Level 4 answers the question, “Is this what the organization exists to
do/deliver/contribute?” (Kirkpatrick Partners, 2015, p. 6). In the case of this study, Level 4
determines the degree to which students are prepared to thrive in a changing world. Measuring
this broad goal is difficult as the goal requires the effort of many in the organization. To help
ensure that the organization is on track to positively influence the desired results, leading
indicators are outlined. These are short-term observations and measurements of individual
initiatives and efforts in support of the overall desired results of the organization.
Table 14 shows the proposed outcomes, metrics, and methods used to evaluate the results
and leading indicators of Level 4.

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Table 14
Outcomes, Metrics, and Methods for Internal Outcomes
Outcome Metric Method
Internal outcomes
All students will have
deep mathematical
understanding with a
focus on content and
21st century skills
Percentage of students scoring
above 80%
Grade level unit assessments
Students feel successful in
mathematics
Score on student identity
measure
Math identity survey

Level 3: Behavior
Level 3 of the new world Kirkpatrick model measures the degree to which participants
have applied what they learned when they are back in their operational setting, the classroom in
this study. Level 3 consists of critical behaviors and required drivers, which are described below.
Critical behaviors. Critical behaviors are a few, specific actions that will have the
biggest impact on Level 4 performance if done reliably. These critical behaviors must be
specific, observable, and measurable. Table 15 below outlines the three critical behaviors for this
plan.

Table 15
Critical Behaviors, Metrics, Methods, and Timing for Evaluation
Critical Behavior Metric Method Timing
1. Teachers will create
an environment
where all children
a. Structures and
protocols are used
a. Observations by
instructional coach,
Beginning, middle,
and end of the
year.

134
Critical Behavior Metric Method Timing
are expected to
participate, where
all voices are heard,
and the authority
resides in the voices
of children.
to ensure all
children participate.
administrators, and
math leaders.
b. Teachers will be
asked to self-report on
the structures and
protocols used in the
classroom to
encourage all students
to participate, and
how they feel the
structures and
protocols have
supported all children
to feel like they are
powerful contributors
in the math classroom.
b. Teachers press
children to
elaborate on, or
clarify, their
thinking.
Observations by
instructional coach,
administrators, and
math leaders.
Beginning, middle,
and end of the
year.

c. Teachers utilize
children’s thinking
to move the
conversation
forward.
Observations by
instructional coach,
administrators, and
math leaders.
Beginning, middle,
and end of the
year.
d. Teachers orient
students towards
others’ thinking to
create shared
meaning.
Observations by
instructional coach,
administrators, and
math leaders.
Beginning, middle,
and end of the
year.
e. In whole group
discussion,
selection of ideas is
strategic and there
is purposeful
sequencing to
support the
mathematical focus
of the lesson.
a. Observations by
instructional coach,
administrators, and
math leaders.
b. Teachers will submit
a planning sheet that
shows how they
planned for the
summary of a lesson
and explain why they
selected and
sequenced the
Beginning, middle,
and end of the
year.

135
Critical Behavior Metric Method Timing
strategies the way
they did.
2. Teachers have high
expectations for all
children.
a. No bias is present
in the interactions
between teachers and
children. This
includes verbal and
non-verbal behaviors.
b. When teachers are
supporting children,
they do not reduce the
level of cognitive
demand.
a. Observations by
instructional coach,
administrators, and math
leaders.
Beginning, middle,
and end of the
year.
3. Teachers will
utilize a variety of
tasks that support
deep mathematical
learning for all.
a. Tasks are utilized
that are rich,
appropriately
challenging, complex,
and lend to multiple
entry points and
solution pathways.
b. Tasks are utilized
that are authentic,
build on students’
experiences, and
connect mathematics
to their lives.
c. Tasks are utilized
that provide
opportunities for
students to
understand, critique,
and change inequities
or social justice issues
within the world with
mathematics.
a. Curriculum Audit by
grade level math leaders
b. Observations by
instructional coach,
administrators, and math
leaders.
a. Examine two
units from the new
curricular resource
per grade level once
throughout the year.
b. Beginning,
middle, and end of
the year.

Required Drivers. Required drivers are the processes and systems which support the
enactment of the critical behaviors. Kirkpatrick and Kirkpatrick (2016) indicated that “active

136
execution and monitoring of required drivers is perhaps the biggest indicator of program success
for any initiative” (p. 7). In this study, several required drivers have been identified for the
successful enactment of EMTPs in order to reinforce, encourage, reward, and monitor the critical
behaviors. Refer to Table 16 for details.

137
Table 16
Required Drivers to Support Critical Behaviors
Method Timing Critical behavior supported
Reinforcing
A collaborative committee
will create a mathematics
vision with clear indicators
of success for enacting
EMTPs.
Beginning of the school year 1, 2, 3
A look-for tool that outlines
what the EMTPs look like
in action will be devised
and shared with teachers.
Beginning of the school year 1, 2, 3
Administrators, coaches, and
math leaders will calibrate
understandings of the look-
for tool to ensure consistent
enactment and feedback.
Quarterly 1, 2, 3
A coherent and sustained
professional learning plan
based on the vision, which
includes training on how to
implement the curricular
materials in accordance
with the vision will be
developed by the
instructional coaches and
administrators.
Beginning of the school year 1, 2, 3
High-quality curricular
materials that align with the
standards and the
organization’s vision will
be selected.
Beginning of the school year 3
Encouraging
Administration, in
conjunction with the
Central Office, will ensure
there is dedicated, focused,
and facilitated PLC time to
collaboratively learn and
reflect on math teaching
practices. Discussions
Bi-weekly 1, 2, 3

138
Method Timing Critical behavior supported
should focus on
implementation and
adjustment of the curricular
resource and the impact on
student learning as well as
how EMTPs support
opportunities for deep math
learning for all.
Discussions should be
aligned to the vision and
look-for tool.
Lab sites will be organized by
the instructional coach to
see EMTPs modeled.
Quarterly 1, 2, 3
Instructional coaches will
provide opportunities for
teachers to engage as
students in lessons that
utilize EMTPs.
Quarterly 1, 2, 3
Teachers will practice new
skills in the classroom with
coaches providing coaching
and feedback.
Following professional
learning sessions
1, 2
Teachers will set and monitor
goals related to the EMTPs.
Quarterly 1, 2, 3
Rewarding
Student identity and learning
success in PLCs based on
anecdotal and summative
and formative assessments
will be shared in PLCs.
Weekly in PLCs 1, 2, 3
MAP data will be examined
for growth.
Beginning and ending of
school year
1, 2, 3
Video or anecdotal examples
of EMTPs in action will be
shared by administrators.
Quarterly at faculty meetings 1, 2, 3
Monitoring
Goal reflection to monitor
growth will take place in
PLCs.
Quarterly

139
Method Timing Critical behavior supported
Learning walks with
administrators, coaches and
math leaders using the look-
for tool to occur.
Quarterly

Organizational Support. Organizations must support the stakeholders’ critical behaviors
if goals are to be met. At GVRIS, there is a lack of clarity of what EMTPs are, how they support
opportunities for all students to engage in deep math learning, and how to implement them
effectively. For teachers to know, understand, and enact EMTPs, there are several ways that
GVRIS can support the critical behaviors. To begin, clarity needs to be created around what
EMTPs are through a collaboratively developed and shared vision. This vision will serve as a
framework to ensure that critical behaviors are enacted successfully and consistently. Further,
curricular resources need to be selected that align to the vision. A coherent and sustainable
professional learning plan which includes opportunities for learning about EMTPs, how to enact
them, and how the resource can be utilized to support these behaviors also needs to be
developed.
Level 2: Learning
Level 2 learning evaluation seeks to understand the degree to which participants have
obtained the knowledge, skills, and attitudes that were intended by the program. Level 2
evaluation includes knowledge and skill, attitude, confidence, and commitment.
Learning Goals. Chapter 4 highlighted several high-priority knowledge, motivation, and
organizational solutions to support the enactment of EMTPs. Critical behaviors have been

140
indicated above to support this goal. To perform the critical behaviors, teachers must know and
understand or be able to do the following:
• Know and understand goals for mathematics and math teaching
• Know and understand what equitable math teaching practices are and how they
promote equitable opportunities for learning
• Know how to enact equitable math teaching practices in their classrooms
Program. The goals listed will be achieved through a year-long professional learning
program for teachers, coaches, and administrators, coordinated or facilitated by a coach. Part of
the program will include in-classroom support with coaching and feedback and focused time and
support during PLCs. Administrators will ensure this is possible by minimizing initiatives and
preserving dedicated time for PLCs to focus on mathematics.
While different stakeholders have different roles in implementation, all stakeholders will
participate in professional development sessions to ensure alignment in understanding for
consistency in enactment and feedback, and to build a culture of collaborative learning. The
calibration and use of the look-for tool will support consistency. Professional development
courses will take place quarterly and will be followed by coaching and feedback throughout the
quarter. Each teacher will be provided with feedback or coaching at least once by a coach or
administrator during the quarter.
Many strategies will be used to facilitate learning during professional development and
implementation of knowledge and skills. These include graphic organizers, visual aids with
concrete examples, opportunities to engage in reflection and discussion about important ideas,
teaching practices, student learning, as well as opportunities to connect new learning to personal
experiences. Opportunities to connect theory to practice through seeing EMTPs in action by the

141
during lab sites and practice lessons where teachers act as students will also occur with the coach
facilitating. A coach will provide support using the curricular resource within professional
development and coaching sessions as well as PLCs.
Evaluation of the components of learning. To determine effectiveness of the program,
checks for declarative knowledge, procedural skills, attitude, confidence, and commitment are
necessary. If one piece is missing, the critical behaviors are less likely to be performed
successfully. Table 17 describe the methods or activities for the evaluation of each of the
components for learning along with timing indicators.

Table 17
Evaluation of the Components of Learning for the Program
Method or activity Timing
Declarative knowledge “I know it.”
Pre- and post-quizzes of knowledge and
understanding checks of the appropriate
professional development sessions.
Before and after professional development
session
Pre- and post-quizzes of knowledge and
understanding checks for goals and purposes
of mathematics, what EMTPs are and how
they support equity.
Beginning and end of year
Procedural skills “I can do it right now.”
Observations of teachers’ application of
knowledge of EMTPs in the classroom
Quarterly learning walks
Quarterly observations during coaching cycles
following professional development
Teacher survey At the end of each professional development
session
Pre-and post-quizzes - enacting EMTPs Beginning and end of year
Attitude “I believe this is worthwhile.”
Discussions during PLCs about the importance
of EMTPs to support learning of all students.
Ongoing during PLCs
Discussions with individual teachers during
coaching sessions
During coaching sessions

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Method or activity Timing
Teacher survey At the end of each professional development
session
Confidence “I think I can do it on the job.”
Teacher survey related to confidence of learned
knowledge or skill
At the end of each professional development
session
Teacher survey related to confidence for
individual enactment of EMTPs and
collective efficacy
Quarterly to monitor progress
Informal discussions will teachers after
teaching observation
After classroom visits
Commitment “I will do it on the job.”
Teacher survey related to commitment of
applying the learned knowledge or skill
At the end of each professional development
session and beginning and end of the year
Goal setting and monitoring using the look-for
tool
Quarterly
Observation of teachers during learning walks
and coaching sessions
Quarterly learning walks

Level 1: Reaction
Level 1 evaluation seeks to measure reactions to the program through the areas of
engagement, relevance, and customer satisfaction. Table 18 outlines methods or tools for
evaluating these reactions along with timing of each evaluation.

143
Table 18
Components to Measure Reactions to the Program
Method or tool Timing
Engagement
Course evaluation At the end of each professional development
session
Observations by facilitator during professional
development sessions that indicate initiative
to engage with the material.
Throughout professional development sessions
Observations by facilitator during PLCs that
indicate initiative to engage with the
material.
Throughout PLCs
Relevance
Observation of practice in the classroom Quarterly
Course evaluation At the end of each professional development
session
Evaluation of relevance of practices and
knowledge learned
Quarterly
Customer satisfaction
Evaluation of coaching sessions - teacher
survey
Quarterly
Evaluation of professional development
sessions
Quarterly
Course evaluation At the end of each professional development
session

Evaluation Tools
Immediately Following the Program Implementation
In this evaluation plan, some Level 1 and 2 indicators are assessed directly after program
implementation. A teacher course evaluation is utilized to gauge Level 1 measures including
customer satisfaction, engagement, and relevance directly following each professional learning
the session. Questions ask how satisfied participants are with the experience, how applicable the

144
session is to their classroom teaching, and how engaged they felt during the session using a 4-
point Likert-type scale. A teacher survey is utilized to measure Level 2 indicators including
declarative and procedural knowledge required for enacting EMTPs as well as attitude,
confidence, and commitment for engaging in these behaviors. A variety of assessment types are
used for Level 2 measurement.
Delayed for a Period After the Program Implementation
Kirkpatrick and Kirkpatrick (2016) suggested additional evaluation measures after the
required drivers are engaged and after stakeholders have an opportunity to implement the
knowledge and skills learned. A variety of assessment types have been utilized to measure
Levels 1–4. These include unit assessments, open-ended survey questions, Likert and Likert-type
scale survey questions, multiple choice items, and curriculum analysis.
Data Analysis and Reporting
The data analysis and reporting plan has been informed by Kirkpatrick and Kirkpatrick
(2016). Qualitative and quantitative data will be analyzed throughout the implementation of the
program to provide timely performance updates. A data dashboard, a dynamic tool to display
current program status data, will be utilized so all stakeholders can visualize program progress.
The data dashboard will focus on Levels 3 and 4, specifically student progress on unit
assessment, leading indicators, and performance levels of critical behaviors and required drivers.
To balance qualitative and quantitative data sources and to provide interest, quotes or
testimonials as well as photos and videos will be included as a supplement to the data dashboard.
A final report will also be created to demonstrate the value of the training program. It will
include key evidence for Levels 3 and 4 as well as barriers, how they were resolved, and if they

145
were not resolved, how they could be in the future. For interest, anecdotes will be included like
what will be included in the ongoing data report.
Summary of the Implementation and Evaluation
In the previous sections of this chapter, an integrated implementation and evaluation plan
based on the new world Kirkpatrick model was utilized to provide a framework for executing
and evaluating the recommended solutions to knowledge, motivation, and organization gaps
(Kirkpatrick & Kirkpatrick, 2016). The program was designed using a backwards design
approach, starting with Level 4, results, and Level 3, critical behaviors of primary stakeholders.
This was followed by Level 2, learning, and Level 1, reaction. Together these four levels offer a
path forward towards successfully implementing and evaluating solutions to key problems in an
organization. The suggested plan offered a variety of ideas for ways to support teachers to gain
the knowledge, skills, and motivation needed to enact EMTPs so that all students can develop
deep mathematical knowledge, and to measure progress towards closing the identified gaps.
It has been suggested by Kirkpatrick and Kirkpatrick (2016) that data is collected
throughout the program journey to ensure progress is made and attainment of the leading
indicators is successful. Tables 5–9 detailed critical behaviors and required drivers important for
attaining leading indicators as well as ways to evaluate learning and reactions to that learning
throughout the implementation process.
It is important to know what success will look like from the outset and answer the
question, “Was it worth it?” at the end (Kirkpatrick & Kirkpatrick, 2016). Having a clear
implementation and evaluation plan allows organizations to answer this question and improves
the likelihood of success. With the previous detailed integrated implementation plan and
evaluation plan, the chances of success for all elementary teachers to have the knowledge and

146
motivation to enact EMTPs so that all students develop deep mathematical understanding and
see themselves as knowers and dowers of mathematics, is high.
Limitations and Delimitations
Chapter 3 outlined several limitations and delimitations of this study. These included
limitations relating to time, observations, interviews, and literature. Short time frames to collect
adequate data only allowed for limited observations. Further, observational limitations such as
not being able to observe all behaviors at once or making judgements on presence of bias were
noted. Further, while it is possible to mitigate some caution on part of the participants by
reminding them that I am wearing the hat of a researcher, it is impossible to know if teachers
shared opinions freely because they know that the researcher is passionate about mathematics
and is a mathematics leader. Literature bias was another limitation that was indicated in Chapter
3.
Delimitations of the study in terms of selection criteria and process were noted in Chapter
3 and led to limitations of the study. A delimitation of the study from the outset was the selection
criteria, which included teachers in grades 3–5, teachers who have been at the school for less
than 3 years, and classroom teachers only. To increase a variety of perspectives, it would have
been useful to include new teachers, primary years teachers, and support staff. Another limitation
of the study was realized when it was noted that criteria for selection did not delineate between
math leaders and non-math leaders. Math leaders have more opportunities for engaging in
professional learning and the utilization of a piloted resource this year. It was difficult to
determine what impacted the knowledge and motivation of math leaders as compared to those
who were not math leaders.

147
An additional limitation of the study resulted from the professional development that took
place during the school year. During the timeframe of the study, some professional development
was provided to all teams in the elementary school either by math leaders or the instructional
coach. This may have skewed the findings of the study.
Limitations also included impacts of COVID-19. The nature of COVID-19 restrictions
may have impacted teachers' ability to enact EMTPs as children were allowed only to be in small
groups seated at their tables. Children were not able to convene at a common meeting space,
which helps to facilitate whole-class discussion and collective meaning-making. Additionally,
physical tools could not be shared so limited tools were available. Finally, COVID-19 has
resulted in extreme teacher burnout as has been shared by teachers anecdotally. This may have
impacted the researcher’s ability to acquire participants for the study and the quality of those
participants' teaching who volunteered to participate in the study.
Recommendations for Future Research
Several themes emerged during the study that would benefit from further examination.
To begin, instructional assistants were not included as stakeholders because they are not the
primary stakeholders responsible for teaching and learning of mathematics. However, GVRIS
has a plethora of instructional assistants that support instruction in the classroom. Further study
would include the role of instructional assistants in supporting teachers in the enactment of
EMTPs and opportunities for knowledge and motivation building to feel successful in providing
said support.
Another idea worthy of exploration is building fluency from conceptual understanding.
Fluency was not considered as an assumed knowledge influence in this study but emerged in the
findings. Most participants provided practice opportunities to develop fluency. It should be noted

148
that while several of the considered EMTPs were not present in Participant 1’s classroom,
children had reportedly engaged in several days of lessons to make sense of the concepts needed
for the procedure. During the two observations, Participant 1 was supporting students to solidify
the standard algorithm as indicated in the grade level resource, which is not aligned to standards.
Two questions to be considered for further research in terms of fluency are:
1. At what point do children have enough conceptual understanding to move into
procedural fluency practice?
2. If there are strategies that children must use according to the standards, how do we
ensure that students are making sense and meeting the standards simultaneously,
especially if time does not permit extended opportunities for development?
Further research should also be undertaken to explore the impact of verbal and non-
verbal communication within the mathematics classrooms and how this impacts students’
mathematical identities, how having a math lab where students are pulled out for higher-level
instruction impacts students’ mathematical identities and teachers’ perceptions of children, and
the role of leadership in successfully leading math programs.
Conclusion
Ensuring that all children in every classroom across the world have opportunities to
engage in deep mathematical learning is important for our future. We must prepare children to
value, appreciate, and use mathematics, and see themselves as knowers and doers of
mathematics. Only powerful classrooms where teachers consistently and successfully enact
equitable mathematics practices will produce these results.
At GVRIS, inequitable opportunities for all children to engage in deep mathematical
learning in classrooms across Grades 3–5 were revealed in the Preschool–12 math review a year

149
prior to the commencement of the study. In depth research of the relevant literature uncovered
possible reasons for the inequities. An adaptation of Clark and Estes’ (2008) KMO framework
was utilized to outline these potential causes through identifying knowledge, motivation, and
organizational influences required to enact EMTPs.
Data to determine knowledge and motivation gaps, as well as organizational supports and
barriers, was collected through semi-structured interviews and classroom observations. Findings
of this study revealed several gaps in knowledge and motivation and highlighted organizational
barriers hindering teachers’ ability to ensure all students engage in deep mathematical learning
resulting in positive mathematical identities.
An integrated implementation and evaluation plan utilizing the new world Kirkpatrick
model (2016) was developed to support progress towards the organizational goal that all children
develop deep mathematical learning. Features of this model outlined recommended solutions and
measurement tools to ensure that teachers have the support needed to realize their goal of
enacting EMTPs.
It is with great excitement and hope that GVRIS embarks on this journey to realize its
aspirations by providing all students with the opportunities they deserve in mathematics.
Through impacting the knowledge and motivation of teachers, key players in support of
children’s learning, success is just on the horizon.

150
References
Aguinis, H., & Kraiger, K. (2009). Benefits of training and development for individuals and
teams, organizations, and society. Annual Review of Psychology, 60, 451–474. http://doi:
10.1146/annurev.psych.60.110707.163505
Aguirre, J., Mayfield-Ingram, K., and Martin, D.B. (2013). The impact of identity in K–8
mathematics: Rethinking equity-based practices. NCTM.
Aguirre, J. M., & del Rosario Zavala, M. (2013). Making culturally responsive mathematics
teaching explicit: A lesson analysis tool. Pedagogies, 8(2), 163–190.
https://doi.org/10.1080/1554480X.2013.768518
Allen, K.C. (2011). Mathematics as thinking. A response to "democracy and school math".
Democracy and Education, 19(2), 1–7.
Ambrose, S.A.Bridges, M. W., DiPietro, M., Lovett, M.C., & Norman, M.K., (2010). How
learning works: Seven research-based principles for smart teaching. Jossey-Bass.
Anderson, L., & Krathwohl, D. (2001). A taxonomy for learning, teaching, and assessing (1st
ed.). Longman.
Anthony, G., Hunter, R., Hunter, J., & Duncan, S. (2015). How ambitious is “ambitious
mathematics teaching”? Set: Research Information for Teachers, 2, 45–52.
Atkins, L. & Duckworth, V. (2019). Socially just research as ethical endeavor. In Strunk, K. K.
& Locke, L. A. (Eds.), Research methods for social justice and equity in education (pp.
67–89). Bloomsbury.
Ball, D. L., & Cohen, D. K. (1996). Reform by the book: What is—or might be—the role of
curriculum materials in teacher learning and instructional reform? Educational
Researcher, 25(9), 6–14.

151
Bartell, T., Wager, A., Edwards, A., Battey, D., Foote, M., & Spencer, J. (2017). Toward a
framework for research linking equitable teaching with the standards for mathematical
practice. Journal for Research of Mathematics Education, 48(1), 7–21.
Berry, R. Q. (2019, May). Examining equitable teaching using the mathematics teaching
framework: President’s messages blog. NCTM. https://www.nctm.org/News-and-
Calendar/Messages-from-the-President/Archive/Robert-Q_-Berry-III/Examining-
Equitable-Teaching-Using-the-Mathematics-Teaching-Framework/
Boaler, J. (2002). Learning from teaching: Exploring the relationship between reform curriculum
and equity. Journal for Research in Mathematics Education, 33(4), 239–258.
Boaler, J. (2008). What’s math got to do with it? How parents and teachers can help children
learn to love their least favorite subject. Penguin Group.
Boaler, J. (2015). Mathematical mindsets: Unleashing students′ potential through creative math,
inspiring messages and innovative teaching. Jossey-Bass.
Bourdieu, P., & Wacquant, L. (1992). An invitation to reflexive sociology. University of Chicago
Press.
Broussard, C. A., & Joseph, A. L. (1998). Tracking: A form of educational neglect? Children &
Schools, 20(2), 110–120. https://doi.org/10.1093/cs/20.2.110
Burdman, P. (2018). The mathematics of opportunity: Rethinking the role of math in educational
equity. Just Equations.
California Department of Education, Instructional Quality Commission, & California State
Board of Education. (2021). Mathematics framework for California Public Schools:
Kindergarten through grade twelve. https://www.cde.ca.gov/ci/ma/cf/

152
Celedón-Pattichis, S., Peters, S., Borden, L., Males, J., Pape, S., Chapman, O., Clements, D., &
Leonard, J. (2018). Asset-based approaches to equitable mathematics. Journal for
Research in Mathematics Education, 49(4), 373–389.
https://doi.org/10.5951/jresematheduc.49.4.0373
Chai, D. S., Hwang, S. J., & Joo, B. K. (2017). Transformational leadership and organizational
commitment in teams: The mediating roles of shared vision and team-goal commitment.
Performance Improvement Quarterly, 30(2), 137–158. https://doi.org/10.1002/piq.21244
Charalambous, C. Y., & Hill, H. C. (2012). Teacher knowledge, curriculum materials, and
quality of instruction: Unpacking a complex relationship. Journal of Curriculum Studies,
44(4), 443–466. https://doi.org/10.1080/00220272.2011.650215
Chong, W. H., & Kong, C. A. (2012). Teacher collaborative learning and teacher self-efficacy.
The Journal of Experimental Education, 80(3), 263–283.
https://doi.org/10.2307/26594354
Clark, L. M., Depiper, J. N., Frank, T. J., Nishio, M., Campbell, P. F., Smith, T. M., Griffin, M.
J., Rust, A. H., Conant, D. L., & Choi, Y. (2014). Teacher characteristics associated with
mathematics teachers’ beliefs and awareness of their students’ mathematical dispositions.
Journal for Research in Mathematics Education, 45(2), 246–284.
Clark, R. E., & Estes, F. (2008). Turning research into results: A guide to selecting the right
performance solutions. CEP Press.
Cobb, P., and Hodge, L. (2002). A relational perspective on issues of cultural diversity and
equity as they play out in the mathematics classroom. Mathematical Thinking and
Learning, 4(2), 249–284.

153
Copur-Gencturk, Y. (2012). Teachers’ mathematical knowledge for teaching, instructional
practices, and student outcomes. [Unpublished doctoral dissertation]. Graduate College
of the University of Illinois at Urbana-Champaign.
Copur-Gencturk, Y., & Thacker, I. (2021). A comparison of perceived and observed learning
from professional development: Relationships among self-reports, direct assessments,
and teacher characteristics. Journal of Teacher Education, 72(2), 138–151.
https://link.gale.com/apps/doc/A653470253/AONE?u=usocal_main&sid=AONE&xid=ef
e8e706
Crosby, M., & Owens, E. (1993). The disadvantages of tracking and ability grouping: A look at
cooperative learning. (Solutions and Strategies Series, Number 5). National Dropout
Prevention Center. http://dropoutprevention.org/wp-content/uploads/2015/08/SS05.pdf
Darling-Hammond, L., Wei, R. C., Andree, A., Richardson, N., & Orphanos, S. (2009).
Professional learning in the learning profession: A status report on teacher development
in the United States and abroad. National Staff Development Council.
https://edpolicy.stanford.edu/sites/default/files/publications/professional-learning-
learning-profession-status-report-teacher-development-us-and-abroad.pdf
Dietiker, L., Males, L. M., Amador, J. M., & Earnest, D. (2018). Curricular noticing: A
framework to describe teachers’ interactions with curriculum materials. Journal for
Research in Mathematics Education, 49(5), 521–532.
https://doi.org/10.5951/jresematheduc. 49.5.0521
Denler, H., Wolters, C., & Benzon, M. (2011). Social cognitive theory.
https://www.researchomatic.com/Social-Cognitive-Theory-86283.html
Dweck, C.S. (2006). Mindset: The new psychology of success. Ballantine Books.

154
English, L. D., & Gainsburg, J. (2015). Problem solving in a 21st-century mathematics
curriculum. In L.D. English & D. Kirshner (Eds.), Handbook of international research in
mathematics education (3rd ed., pp. 313–335). Routledge.
https://doi.org/10.4324/9780203448946
Enochs, L. G., Smith, P. L., Huinker, D. (2000). Establishing factorial validity of the
Mathematics Teaching efficacy Beliefs Instrument. School Science and Mathematics,
100(4), 194–202. https://doi.org/10.1111/j.1949-8594.2000.tb17256.x
Ernest, P. (1991). The philosophy of mathematics education. (1st ed.). Taylor & Francis.
Ertmer, P. A., & Simons, K. D. (2005). Scaffolding teachers’ efforts to implement problem-
based learning. International Journal of Learning, 12(4), 319–328.
Every Student Succeeds Act, 20 U.S.C. § 6301 (2015). https://www.ed.gov/essa?src=rn
Felton-Koestler, M. D. (2019). Children know more than I think they do: The evolution of one
teacher’s views about equitable mathematics teaching. Journal of Mathematics Teacher
Education, 22(2), 153–177. https://doi.org/10.1007/s10857-017-9384-0
Flores, A. (2007). Examining disparities in mathematics education: Achievement gap or
opportunity gap? The High School Journal, 91(1), 29–42.
Gallimore & Goldenberg. (2001). Analyzing cultural models and settings to connect minority
achievement and school improvement research. Educational Psychologist 36(1), 45–56.
Garet, M. S., Porter, A. C., Desimone, L., Birman, B. F., & Yoon, K. S. (2001). What makes
professional development effective? Results from a national sample of teachers.
American Educational Research Journal, 38(4), 915–945.
Gasser, K. W. (2011). Five ideas for 21st century math classrooms. American Secondary
Education, 39(3), 108–116. https://www.jstor.org/stable/23100426

155
Gay, G. (2002). Preparing for culturally responsive teaching. Journal of Teacher Education,
53(2), 106–116. https://doi.org/10.1177/0022487102053002003
Gay, G., (2010). Culturally responsive teaching: Theory, research, and practice (2nd ed.).
Teachers College Press.
Gibbons, L., & Cobb, P. (2016). Content-focused coaching: Five key practices. The Elementary
School Journal, 117(2), 237–260. https://doi.org/10.1086/688906
Glesne, C. (2011). Chapter 6: But is it ethical? Considering what is right. Becoming qualitative
researchers: An introduction, 4, 162–183.
Goddard, R.D., Hoy, W. K., & Hoy, A. W. (2000). Collective teacher efficacy: Its meaning,
measure, and impact on student achievement. American Educational Research Journal,
37, 479–507.
Gutiérrez, R. (2002). Enabling the practice of mathematics teachers in context: Toward a new
equity research agenda. Mathematical Thinking and Learning, 4(2–3), 145–187.
https://doi.org/10.1207/s15327833mtl04023_4
Gutiérrez, K., Rymes, B., & Larson, J. (1995). Script, counterscript, and underlife in the
classroom: James Brown versus Brown v. Board of Education. Harvard Educational
Review, 65, 444–471.
Gutiérrez, R., & Irving, S. E. (2012). Latino/a and Black students and mathematics. The Students
at the Center Series. Jobs for the Future. https://files.eric.ed.gov/fulltext/ED561299.pdf
Gutstein, E. (2012). Reading and writing the world with mathematics: Toward a pedagogy for
social justice. Taylor & Francis. https://doi.org/10.4324/9780203112946
Hallinan, M. T. (1994). Tracking: From theory to practice. Sociology of Education, 67(2), 79–84.

156
Harris, A., Jones, M., & Huffman, J. B. (Eds.). (2017). Teachers leading educational reform:
The power of professional learning communities. Routledge.
Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., & Ball, D.
L. (2008). Mathematical knowledge for teaching and the mathematical quality of
instruction: An exploratory study. Cognition and Instruction, 26(4), 430–511.
https://doi.org/10.1080/07370000802177235
Hill, H. C., Charalambous, C. Y., & Chin, M. J. (2019). Teacher characteristics and student
learning in mathematics: A comprehensive assessment. Educational Policy, 33(7), 1103–
1134. https://doi.org/10.1177/0895904818755468
Huinker, D., & Bill, V. (2017). Taking action: Implementing effective mathematics teaching
practices in grades k–grade 5. NCTM.
Jackson, K. & Cobb, P. (2013). Coordinating professional development across contexts and role
groups. In M. Evans (Ed.), Teacher education and pedagogy: Theory, policy, and
practice (pp. 80–99). Cambridge University Press.
Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s
mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169 –202.
Kirkpatrick, D. L. (2006). Seven keys to unlock the four levels of evaluation. Performance
Improvement, 45(7) 5–8.
Kirkpatrick, K. & Kirkpatrick, W. (2015). An Introduction to the new world Kirkpatrick model.
Kirkpatrick Partners.
Kirkpatrick, K. & Kirkpatrick, W. (2016). Kirkpatrick’s four levels of training evaluation.
Associated Press.

157
Kirschner, P., Kirschner, F., & Paas, F. (2009). A Cognitive load approach to collaborative
learning: United brains for complex tasks. Educational Psychology Review, 21, 31-42.
Kennedy, M. M. (2016). How does professional development improve teaching? Review of
Educational Research, 86(4), 945–980. https://doi.org/10.3102/0034654315626800
Kleickmann, T., Richter, D., Kunter, M., Elsner, J., Besser, M., Krauss, S., & Baumert, J. (2013).
Teachers’ content knowledge and pedagogical content knowledge: The role of structural
differences in teacher education. Journal of Teacher Education, 64(1), 90–106.
https://doi.org/10.1177/0022487112460398
Kotter, J. P. (2011). Leading change: Why transformation efforts fail. In J.P. Kotter, W.C. Kim,
& R.A. Mauborhne (Eds.), On change management. (pp. 1–16). Harvard Business
Review Press.
Ladson-Billings, G. (1997). It doesn't add up: African American students' mathematics
achievement. Journal for Research in Mathematics Education, 28(6), 697–708.
Ladson-Billings, G. (2009). The Dreamkeepers: Successful teachers of African American
children. Jossey-Bass.
Lampert, M., Boerst, T., & Graziani, F. (2011). Organizational resources in the service of
school-wide ambitious teaching practice. Teachers College Record, 113(7), 1361–1400.
Learning Mathematics for Teaching/Heather Hill. (2014). Mathematical quality of instruction
(MQI): 4-point version. https://cepr.harvard.edu/mqi
Lin, H., Hong, Z., Yang, K., & Lee, S. T. (2013). The impact of collaborative reflections on
teachers’ inquiry teaching. International Journal of Science Education, 35(18), 3095–
3116. https://doi.org/10.1080/09500693.2012.689023

158
Lochmiller, C.R. & Lester, J. N. (2017). An introduction to educational research: Connecting
methods to practice. Sage.
Loughland, T., & Nguyen, H. T. M. (2020). Using teacher collective efficacy as a conceptual
framework for teacher professional learning: A case study. Australian Journal of
Education, 64(2), 147–160. https://doi.org/10.1177/0004944120908968
Louie, N. L. (2017). The culture of exclusion in mathematics education and its persistence in
equity-oriented teaching. Journal for Research in Mathematics Education, 48(5), 488–
519. https://doi.org/10.5951/jresematheduc.48.5.0488
Louie, N. (2020). Agency discourse and the reproduction of hierarchy in mathematics
instruction. Cognition and Instruction, 38(1), 1–26.
https://doi.org/10.1080/07370008.2019.1677664
Loveless, T. (1998). The tracking and ability grouping debate. Fordham Report, 2(8), 1–3.
https://files.eric.ed.gov/fulltext/ED422454.pdf
Lubienski, S. T. (2000). Problem solving as a means toward mathematics for all: An exploratory
look through a class lens. Journal for Research in Mathematics Education, 31(4), 454–
482. https://doi.org/10.2307/749653
Matthews, L. E., Jones, S. M., & Parker, Y. A. (2013). Advancing a framework for culturally
relevant, cognitively demanding mathematics tasks. In Leonard, J. & Martin, D. (Eds.),
The brilliance of Black children in mathematics: Beyond the numbers and toward new
discourse, (pp. 123-150). Information Age Publishing.
Maxwell, J. A. (2013). Qualitative research design: An interactive approach (3rd ed.). Sage.
Merriam, S. B., & Tisdell, E. J. (2016). Qualitative research: A guide to design and
implementation (4th ed.). Jossey-Bass.

159
Mokros, J., Economopoulos, K., & Russell, S. (1995). Beyond arithmetic: Changing
mathematics in the elementary classroom. Dale Seymour Publications.
Morvan, J. (2017). Making visible and acting on issues of racism and racialization in school
mathematics. Brock Education, 27(1), 35–48. https://doi.org/10.26522/brocked.v27i1.624
Moscardini, L. (2014). Developing equitable elementary mathematics classrooms through
teachers learning about children’s mathematical thinking: Cognitively Guided Instruction
as an inclusive pedagogy. Teaching and Teacher Education, 43, 69–79.
https://doi.org/10.1016/j.tate.2014.06.003
Mueller, C. & Dweck, C. (1998). Praise for intelligence can undermine children’s motivation and
performance. Journal for Personality and Social Psychology, 75(1), 33–52.
Munson, J. (2018). Responding to student thinking in the moment: Examining conferring
practices and teacher learning in the elementary mathematics classroom [Unpublished
doctoral dissertation]. The Graduate School of Education of Stanford University.
http://purl.stanford.edu/nf372fz4033
National Association for the Education of Young Children. (2019). Advancing equity in early
childhood education. https://www.naeyc.org/sites/default/files/globally-
shared/downloads/PDFs/resources/position-
statements/advancingequitypositionstatement.pdf
National Council of Teachers of Mathematics. (2000). Principles and standards for school
mathematics. NCTM.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring
mathematical success for all. NCTM.

160
National Council of Teachers of Mathematics. (2020). Catalyzing change in early childhood and
elementary mathematics. NCTM.
National Council of Supervisors of Mathematics. (2019). NCSM essential actions: Instructional
leadership in mathematics education. NCSM.
National Governors Association Center for Best Practices, Council of Chief State School
Officers. (2010). Common Core State Standards Math. National Governors Association
Center for Best Practices, Council of Chief State School Officers.
http://www.corestandards.org/Math/
National Research Council, & Mathematics Learning Study Committee. (2001). Adding it up:
Helping children learn mathematics. National Academies Press.
No Child Left Behind Act, 20 U.S.C. § 6319 (2001). https://www2.ed.gov/nclb/landing.jhtml
Oakes, J. (1992). Can tracking research inform practice? Technical, normative, and political
considerations. Educational Researcher, 21(4), 12–21.
Patel, L. (2015). Educational research as a site of coloniality. In Decolonizing educational
research: From ownership to answerability (pp. 11–28). Routledge.
https://doi.org/10.4324/9781315658551-2
Patton, M. Q. (2002). Chapter 7: Qualitative Interviewing. In Qualitative research & evaluation
methods (3rd ed.) (pp. 339–380). Sage.
RAND Corporation. (2019). Teachers matter: Understanding teachers’ impact on student
achievement. https://www.rand.org/pubs/research_reports/RR4312.html
Ravitch, S. M., & Carl, N. M. (2012). Conceptual frameworks in research. In Qualitative
research: Bridging the conceptual, theoretical, and methodological. (2nd ed.) (pp. 32–
61). Sage.

161
Remillard, J. T., Reinke, L. T., & Kapoor, R. (2019). What is the point? Examining how
curriculum materials articulate mathematical goals and how teachers steer instruction.
International Journal of Educational Research, 93, 101–117.
https://doi.org/10.1016/j.ijer.2018.09.010
Robinson, S. B., & Firth Leonard, K. (2019). Designing quality survey questions. Sage.
Rosenquist, B. A., Henrick, E. C., & Smith, T. M. (2015). Research–practice partnerships to
support the development of high-quality mathematics instruction for all students. Journal
of Education for Students Placed at Risk, 20(1–2), 42–57.
Rueda, R. (2011). The 3 dimensions of improving student performance. Teachers College Press.
Schlesinger, L., Jentsch, A., Kaiser, G., König, J., & Blömeke, S. (2018). Subject-specific
characteristics of instructional quality in mathematics education. ZDM - Mathematics
Education, 50(3), 475–490. https://doi.org/10.1007/s11858-018-0917-5
Schneider, R. M., & Krajcik, J. (2002). Supporting science teacher learning: The role of
educative curriculum materials. Journal of Science Teacher Education, 13(3), 221–245.
Schunk, D. (2020). Learning theories: An educational perspective (8th ed.). Pearson Education.
Schunk, D. H. & DiBenedetto, M. K. (2016). Self-efficacy theory in education. In K. R. Wentzel
& D. B. Miele (Eds.), Handbook of motivation at school (2nd ed., pp. 34–54). Routledge.
Schunk, D. H., Pintrich, P. R., & Meece, J. L. (2009) Motivation in education: Theory, research,
and application. Pearson/Merrill Prentice Hall.
Seeley, C. L. (2009). Faster isn’t smarter: Messages about math, teaching, and learning in the
21
st
century. Math Solutions.
Strategic Education Research Partnership (2014, September 26). The 5x8 card. SERP Institute.
https://www.serpinstitute.org/5x8-card

162
Sharp, J. G., Hopkin, R., & Lewthwaite, B. (2011). Teacher perceptions of science in the
National Curriculum. International Journal of Science Education, 33, 2407–2436.
Sherin, M. G., & Drake, C. (2009). Curriculum strategy framework: Investigating patterns in
teachers’ use of a reform‐based elementary mathematics curriculum. Journal of
Curriculum Studies, 41(4), 467–500.
Smith, M.S. & Stein, M. K. (2011). 5 practices for orchestrating productive mathematics
discussions. NCTM.
Suh, J., Matson, K., Seshaiyer, P., Jamieson, S., & Tate, H. (2021). Mathematical modeling as a
catalyst for equitable mathematics instruction: Preparing teachers and young learners
with 21st century skills. Mathematics, 9(162), 1–20.
https://doi.org/10.3390/math9020162
Szabo, Z. K., Körtesi, P., Guncaga, J., Szabo, D., & Neag, R. (2020). Examples of problem-
solving strategies in mathematics education supporting the sustainability of 21st-century
skills. Sustainability (Switzerland), 12, 1–28. https://doi.org/10.3390/su122310113
Tan, P., Lambert, R., Padilla, A., & Wieman, R. (2019). A disability studies in mathematics
education review of intellectual disabilities: Directions for future inquiry and practice.
Journal of Mathematical Behavior, 54, 100672.
https://doi.org/10.1016/j.jmathb.2018.09.001
The National Assessment of Educational Progress (2019). The nation’s report card:
Mathematics. https://www.nationsreportcard.gov/mathematics/nation/scores/?grade=4
Thomson, M. M., & Gregory, B. (2013). Elementary teachers’ classroom practices and beliefs in
relation to US science education reform: Reflections from within. International Journal
of Science Education, 35(11), 1800–1823.

163
Toheri, Winarso, W., & Haqq, A. A. (2020). Where exactly for enhance critical and creative
thinking: The use of problem posing or contextual learning. European Journal of
Educational Research, 9(2), 877–887. https://doi.org/10.12973/eu-jer.9.2.877
Turner, M. (2016). Teachers’ practice of mathematical reform techniques in the classroom.
[Unpublished doctoral dissertation]. Walden University.
Van de Walle, J., Karp K., Bay-Williams, J. (2012). Elementary and middle school mathematics:
Teaching developmentally (8th ed.). Pearson.
Villaverde, L. (2008). Feminist primer. Peter Lang.
Voelkel, R. H., & Chrispeels, J. H. (2017). Understanding the link between professional learning
communities and teacher collective efficacy. School Effectiveness and School
Improvement, 28(4), 505–526.
Vygotsky, L. S. (1962). Thought and language. MIT Press.
Wachira, P. & Mburu, J. (2019). Culturally Responsive Mathematics Teaching and
Constructivism: Preparing Teachers for Diverse Classrooms. Multicultural Learning and
Teaching, 14(1), 20160023. https://doi.org/10.1515/mlt-2016-0023
Windschitl, M., Thompson, J., & Braaten, M. (2011). Ambitious pedagogy by novice teachers:
Who benefits from tool-supported collaborative inquiry into practice and why? Teachers
College Record, 113(7), 1311–1360.
Wright, P. (2012). The Math Wars: Tensions in the development of school mathematics
curricula. For the Learning of Mathematics, 32, 7–13.
Yolcu, A. (2019). Research on equitable mathematics teaching practices: Insights into its
divergences and convergences. Review of Education, 7(3), 701–730.
https://doi.org/10.1002/rev3.3163

164
Figures
Figure 1

The Mathematics Teaching Framework

Note. From “Taking Action: Implementing Effective Mathematics Teaching Practices,” by
Huinker, D. and Bill, V., 2017. Copyright 2017 by NCTM.

165

Appendix A: Interview Protocol for Teachers
Name: Gynelle Gaskell
Research Question(s):
1. What is the current status of teachers’ knowledge and motivation related to enacting
equitable mathematics teaching practices, or those teaching practices that provide access
to deep mathematics learning for all learners?
2. How do organizational factors influence teachers’ capacity to enact equitable
mathematics teaching practices?

I. Introduction:
Thank you for agreeing to participate in my study. I really appreciate the time that you have set
aside to answer my questions. As I mentioned when we last spoke, the interview should take
about an hour, does that still work for you?
Before we get started, I want to remind you about this study, the overview for which was
provided to you in the Study Information Sheet, and answer any questions you might have about
participating in this interview. I am a student at USC and am conducting a study to understand
how teachers in Grades 3–5 teach mathematics.
I am particularly interested in understanding how teachers feel supported by the organization to
have the knowledge and motivation to be able to teach mathematics, which is why you were
chosen for this study. I will be talking to several teachers in Grades 3–5 to learn more about
mathematics teaching.
I want to assure you that I am strictly wearing the hat of a researcher today. I recognize that I am
an instructional coach with a background in mathematics and realize my biases towards this
topic. However, I want you to know that I am truly here to learn so that we can make
improvements to our organization. The nature of my questions is not evaluative and there will
not be any judgments made on how you are performing as a teacher. My goal is to understand
your perspective, both positive and negative aspects so that we can make appropriate changes.
As stated in the Study Information Sheet I provided to you previously, this interview is
confidential. What that means is that your name will not be shared with anyone including other
teachers, coaches, the principal, or the Office of Learning. The data for this study will be
compiled into a report and while I do plan on using some of what you say as direct quotes, none
of this data will be directly attributed to you. I will use a pseudonym to protect your
confidentiality and will try my best to de-identify any of the data I gather from you. I am happy
to provide you with a copy of my final paper if you are interested.
As stated in the Study Information Sheet, I will keep the data in a password protected computer
and all data will be destroyed after 3 years.]
Do you have any questions about the study before we get started? I have brought a recorder with
me today so that I can accurately capture what you share with me. The recording is solely for
my purposes to best capture your perspectives and will not be shared with anyone outside the
166

research team except for the transcription service. May I have your permission to record our
conversation? I will also be taking some notes.
Ok, let’s begin.
I. Background Questions
I’d like to start by asking you some background questions to learn a little about you.
1. Tell me about becoming a teacher. What made you decide to be a teacher?
2. How do you feel about teaching? What makes you say this?
3. Describe your math experience growing up.
a. Follow-up: How has this impacted how you teach math, if at all?
4. Tell me about your experiences teaching math.
a. What grade levels have you taught?
b. For how long?
5. How do you feel about teaching math? Why?
a. Can you give specific reasons or examples that led you to feel this way?
II. Heart of the Interview
Now I’d like to ask you some questions about the goals of mathematics.
1. In your opinion, what should the goals of mathematics be?
a. Why do we teach mathematics?
b. What are you trying to accomplish in your math lessons?
2. In your opinion, what does it mean to be successful in mathematics?
Let’s shift gears and talk about planning and teaching mathematics.
3. Please describe how you go about planning a math lesson.
a. What do you consider when planning?
b. In what ways do you consider your children when planning?
If content standards or progressions are not mentioned, ask
i. In what ways do you use content standards for planning, if at all?
ii. In what ways do you use content progressions for planning, if at all?
4. Suppose I was in your classroom for a typical math lesson.
a. What would I see you doing?
b. What would I hear you saying?
c. What would the children be doing?
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d. What would the children be saying?
5. Tell me about your teaching style in math. How would you describe yourself as a math
teacher?
6. Some people say that children need to learn basic math skills before they can solve
problems. Others say children should learn basic math skills through problem-solving.
What would you say to that?
7. Some people say that as long as a student can solve a problem, it is not necessary to
explain or justify their reasoning. What would you say to that?
8. How do you feel about your ability to teach mathematics to all children?
a. Follow-up: What about those children whose first language is not English, who
have a diagnosed disability, or who learn math a different way at home?
9. When you think about those children who excel in mathematics, what do you believe
are the contributing factors?
10. When you think about those children who struggle in mathematics, what do you
believe are the contributing factors?
11. In your opinion, what are the most important factors in ensuring all children succeed in
mathematics?
a. Follow-up if instructional strategies are not mentioned: Do you believe there is
a correlation between your instructional strategies and children’s success in
mathematics?
i. If yes, please describe a time when your instructional strategies
impacted your children’s success in mathematics?
ii. If no, why not?
Now I’d like to ask you about equity in the mathematics classroom.
12. First, I recognize that we have been using the word equity a lot with our DEI work, but
people may have different definitions of equity. In your opinion, what does equity
mean in the classroom?
a. How would you describe an equitable classroom?
13. If someone were to ask you what is meant by equity in mathematics, what would you
say?
a. What causes inequities in mathematics?
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14. In your opinion, what teaching practices support an equitable mathematics classroom?

15. This is a question about student identity. In your opinion, what does it mean to have a
positive math identity?
a. Some people say it’s not important for children to develop a positive math
identity. What are your thoughts?
So far we’ve discussed teaching practices and your experience. Now I’d like to ask some
questions about the school and how they support teachers in being able to enact EMTPs
16. Suppose I was a new teacher to the school and I asked you to describe what SAS
believes is important in teaching and learning of mathematics. What would you say?
17. Describe your math experience as a teacher at SAS.
a. What might be some contributing factors to this?
18. Describe how your school supports your role as an elementary mathematics teacher, if
at all?
a. How has this impacted your practice, if at all?
19. The school expects teachers to work in Professional Learning Communities (PLCs) to
collectively meet the needs of all children. How confident are you that together with
your team, all children in G_, including those of different racial groups, those who are
ELLs, or those with identified learning disabilities, will be successful in mathematics?
Explain why.
20. Please describe how your team supports each other in collectively figuring out how to
best support your children, if at all.
21. Would you please describe what you think the ideal organization for supporting
teachers in teaching mathematics would be like?
22. What would you say is the school's role in creating the ideal organization you just
spoke about?
a. What suggestions do you have for improvement?
For this last set of questions, I’d like to ask you about the content you will be teaching when I
come to observe in your classroom.
23. I will be observing in your classroom on ___ and ___. The unit you will be teaching is
on place value, addition and subtraction.
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a. Please describe the big ideas of Place Value. What do children need to
understand?
b. How does the positioning and repositioning of the digit change its value?
c. How can understanding place value help to solve problems?
i. Rounding?
ii. Comparing?
iii. Addition?
iv. Subtraction?
v. Assess the reasonableness of an answer?
d. 5th Grade - How is decimal PV related to whole number PV?
e. Children are asked to solve a problem such as ________.
i. What strategies might children use?
1. How are they similar or different?
2. How do the strategies work?
ii. What representations might children use?
iii. How are they similar or different?
f. What are the possible misconceptions children might have in solving this
problem?
g. How might you support them in overcoming the misconception?
24. How do children develop the concepts of PV over time?
a. If you need to extend a student beyond your grade level with PV ideas, what
would you do?
25. How do children develop the concepts of addition over time?
a. If you need to extend a student beyond your grade level with addition ideas,
what would you do?
26. How do children develop the concepts of subtraction over time?
a. If you need to extend a student beyond your grade level with subtraction ideas,
what would you do?
That covers everything I wanted to ask. What other insight would you like to share about our
conversation about elementary teachers teaching mathematics, if any? Anything else I should
have asked that I didn’t think to ask?
170

V. Closing Comments:
Thank you so much for sharing your thoughts with me today! I really appreciate your time and
willingness to share. Everything that you have shared is really helpful for my study. If I have
any follow-up questions, can I contact you, and if so, is email ok? My next step is to finish
conducting the rest of the interviews and classroom observations, and have the audio
transcribed. After reviewing the transcript, should I wish to use your words as quotes, I will
seek your permission before doing so. Again, thank you for participating in my study. As a
thank you, please accept this small token of my appreciation (thank you note and SB GC).

Note. Ideas derived from the “Mathematics Teaching Efficacy Beliefs Instrument (MTEBI)” by
Enochs, L., Smith, P. and Huinker, D, 2000. Copyright 2000 by Enochs, et al. Ideas also derived
from “Mathematics Quality of Instruction (MQI)” by Learning Mathematics for
Teaching/Heather Hill, 2014.

Appendix B: Observation Tool
Failure to enact EMTPs Enactment of EMTPs Notes

Teacher utilizes a different structure for the
lesson which does not allow students to
engage deeply with mathematics.

Tasks
Nature of mathematical tasks is uni-
dimensional (e.g., narrows the focus of
one's thinking, may be focused on a
specific set of procedures, and/or does
not support the need for a diverse set of
group members’ skills.

Task may be rich, complex and invite
speculation, but the teacher continually
lowers the cognitive demand of the task
(e.g., heavily scaffolding the task such
that the opportunity for problem-solving
is minimal).

Students do not understand how the task
connects to their lives; tasks are not
authentic.

Teacher utilizes a 3-part lesson
framework that launches the lesson,
allows students to explore, and then
summarized the mathematics
(enactment: construct knowledge and
make sense, discourse, norms of
participation, all students positioned
as capable, attend explicitly to race,
culture, or other identifying
characteristics).

Tasks
Nature of the mathematical tasks is
rich, appropriately challenging,
complex, and lends to multiple entry
points and solution pathways. Tasks
are posed in ways that invite
speculation (enactment: construct
knowledge and make sense,
differentiation through questioning
and scaffolds or pushes, all students
are positioned as capable).

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Failure to enact EMTPs Enactment of EMTPs Notes
Tasks (continued)
Tasks do not provide opportunities
for students to understand, critique,
and change inequities or social
justice issues within the world with
mathematics.

Discourse and norms of participation
Classroom arrangement does not
support collaborative work or
classroom is arranged to support
collaboration, but students are not
interacting.

Teacher provides no or few
opportunities for students to share
their own thinking or work with
their peers.
Tasks (continued)
Tasks are authentic, build on students’
experiences, and connect the mathematics to
their lives (enactment: all students are
positioned as capable, draws of funds of
knowledge, attend explicitly to race, culture,
or other identifying characteristics).

Tasks provide opportunities for students to
understand, critique, and change inequities or
social justice issues within the world with
mathematics (purposes of mathematics;
enactment: attend explicitly to race, culture,
or other identifying characteristics).

Discourse and norms of participation
Teacher purposefully prompts students to talk
about each other's explanations (enactment:
discourse, norms of participation, all students
are positioned as capable, attend explicitly to
race, culture, or other identifying
characteristics).

172

Failure to Enact EMTPs Enactment of EMTPs Notes
Discourse and norms of participation
(continued)
Teacher both asks and answers his or her own
questions. Teacher asks fill-in-the-blank or
low-level questions. Teacher continues to
call on individual students until a student
provides the response they are looking for.

Teacher may or may not gather data during
the investigation but appears to rely on
volunteers (does not purposefully select and
sequence shares).

Students feel uneasy to share thinking or are
embarrassed to admit mistakes.
Students ask to check answers or how to
complete the task. Students may be focused
on procedures or completion.
Nature of questions and prompts does not
necessarily demand stronger argumentation.
Students’ arguments are focused on what they
did but not necessarily why they did what
they did.

Discourse and norms of participation
(continued)
Teacher appears to be purposefully monitoring and
selecting students to share their presentations with
the class (content and progressions; enactment:
discourse, norms of participation, all students are
positioned as capable, attend explicitly to race,
culture, or other identifying characteristics).

Teacher strategically chooses what students share and
there is purposeful sequencing to support the
mathematical focus of the lesson (content and
progressions; enactment: discourse, norms of
participation, all students are positioned as capable,
attend explicitly to race, culture, or other
identifying characteristics).

Teacher appears to have established a protocol/norm
for the learning culture where students are expected
to participate and are positive and supportive
towards each other (enactment: all students are
positioned as capable, norms of participation,
attend explicitly to race, culture, or other
identifying characteristics).

Teacher pushes students to not only explain how to
do something but also expect students to explain
why it works (enactment - all students are
positioned as capable).

173

Failure to enact EMTPs Enactment of EMTPs Notes
Discourse and norms of participation
(continued)
Teacher continually rephrases or revoices
students’ responses.

The final mathematical authority clearly
resides with the teacher. The teacher
demonstrates and justifies correct
methods and solutions. Teacher does all
of the summarizing.

Students do not seem to be engaged or
excited about the mathematics they are
learning.

Discourse and norms of participation (continued)
Teacher pushes students to argue the validity of a
mathematical statement or solution through
reasoning and justifying as well as critique the
reasoning of others (enactment: all students are
positioned as capable).
Teacher pushes students to build on one another
strategies and thinking as well as generate and
defend arguments (enactment: discourse, all
students are positioned as capable).
The authority seems to reside in with students’
reasoning rather than that of the teacher (enactment:
all students are positioned as capable)
Classroom culture seems to have fostered curiosity
and sense-making, which is reflected both in terms
of the questions that students pose to one another
and in the questions that students think about
themselves (purposes of mathematics: joy and
wonder).

174

Failure to enact EMTPs Enactment of EMTPs Notes
Meeting students’ needs
Teacher gives too many hints and/or
answers questions for the students.

Teacher or other students solve the
problem for the student.

Some students are not appropriately
challenged.

Content
Teacher appears to have anticipated
common student misconceptions but
may miss opportunities to surface them
in ways that support a consolidated
understanding of the concepts.

Teacher makes mathematical errors or
lacks content precision in interactions
with students.

Attending to race, culture, and other
identifying characteristics
Teacher interacts differently with students
based on race, culture, or other
characteristics.

Meeting students’ needs
Teacher provides scaffolds and extends students
according to learning progressions and knowledge of
students without reducing the cognitive demand
(content and progressions; enactment: differentiation
through questioning and scaffolds or pushes, all
students are positioned as capable, draws of funds of
knowledge, attends explicitly to race, culture, or other
identifying characteristics).

Content
Teacher anticipates, notes, and fully addresses common
misconceptions (content).

Teacher anticipates, notes, and fully addresses common
misconceptions (content).

When teacher engages with students through
questioning, clarifying, or choosing student solutions
for presentation, mathematical content is correct and
precise. Teacher addresses incorrect solution methods
(content).

Attending to race, culture, and other identifying
characteristics
Bias according to students’ race, culture, or other
characteristics is not evident. Teacher interacts with
all students with equally high expectations
(enactment: attends explicitly to race, culture, or other
identifying characteristics).

175

Failure to enact EMTPs Enactment of EMTPs Notes
Attending to race, culture, and other
identifying characteristics (continued)
Teacher focuses on incorrect use of
language for ELL students rather than
mathematical ideas.

Deficit language is used such as, “You
can’t do it that way. That’s not correct.
Let me show you. I knew this was
going to be too hard for you.”

Attending to race, culture, and other identifying
characteristics (continued)

Teachers uses asset-based language such as, “You
know this. How can you build on that? I know you
can do this” (enactment: attends explicitly to race,
culture, or other identifying characteristics).

Note. This observational tool has incorporated ideas from the “Mathematical Teaching Practices Continuum” by NCSM, 2019,
NCSM. Copyright by NCSM; the “Mathematics Quality of Instruction (MQI)” by Learning Mathematics for Teaching/Heather Hill,
2014. CC-BY-NC-ND 4.0.; the “Framework for Culturally Relevant, Cognitively Demanding Tasks” by Mathews et al., 2013,
Information Age Publishing. Copyright by Information Age Publishing; and Bartell et al’s (2017) “Equitable Mathematics Teaching
Practices” by Bartell et al., 2017, Journal for Research in Mathematics Education. Copyright by National Council of Teachers of
Mathematics.
176

176

Appendix C: Informed Consent/Information Sheet
University of Southern California
Rossier School of Education
3470 Trousdale Pkwy, Los Angeles CA, 90089

The Enactment of Equitable Mathematics Teaching Practices: An adapted gap analysis

You are invited to participate in a research study conducted by Gynelle Gaskell at the University
of Southern California. Research studies include only people who voluntarily choose to take part.
This document explains information about this study. You should ask questions about anything
that is unclear to you.

PURPOSE OF THE STUDY
This research study aims to understand how teachers in Grades 3–5 teach mathematics. I am
particularly interested in understanding the knowledge and motivation teachers have in order to
teach mathematics effectively. Additionally, I am interested in understanding any organizational
structures teachers may feel support or hinder their ability to teach mathematics effectively.

PARTICIPANT INVOLVEMENT
If you agree to take part in this study, you will be asked to participate in one interview that will
take approximately one hour to complete. You will also be asked to participate in two
observations of approximately 45 minutes over a 2-week time frame. Each interview will be
audio recorded and each observation will be video recorded so I can accurately capture all
information. Audio recordings will be uploaded to an external source for transcript creation.
Questions and observations will be based on the knowledge, motivation, and organizational
influences that impact Grades 3–5 teachers in teaching mathematics. You do not have to answer
any questions you do not wish to.

CONFIDENTIALITY
This study is confidential. Your name will not be shared with administrators, other coaches, or
the Central Office. I will do my best to de-identify identifiable information that is collected. I
will use specific quotes in my report but will ask specific permission from you should I choose
one of your quotes before proceeding. If given permission, I will use a pseudonym to protect
your identity. All data will be kept in a password protected computer and data will be destroyed
after 3 years. A copy of my final report will be provided to you, if interested.

Required language:
The members of the research team, the funding agency and the University of Southern
California’s Human Subjects Protection Program (HSPP) may access the data. The HSPP
reviews and monitors research studies to protect the rights and welfare of research subjects.

When the results of the research are published or discussed in conferences, no identifiable
information will be used.

177
INVESTIGATOR CONTACT INFORMATION
The Principal Investigator is GYNELLE GASKELL, GGAKELL@USC.EDU, 9007 3191
The Faculty Advisors are LAWRENCE PICUS, LPICUS@ROSSIER.USC, DARLINE
ROBLES, DPROBLES@ROSSIER@USC.EDU and YASEMIN COPUR-GENCTURK,
COPURGEN@USC.EDU.

IRB CONTACT INFORMATION
University Park Institutional Review Board (UPIRB), 3720 South Flower Street #301, Los
Angeles, CA 90089-0702, (213) 821-5272 or upirb@usc.edu

178
Appendix D: Recruitment Letter
Dear ________,

As a Doctor of Education student at the University of Southern California, I invite you to
participate in a study to better understand how teachers in Grades 3–5 teach mathematics at the
school. The purpose of this study is to understand the knowledge, motivation, and organizational
factors of teaching mathematics in Grades 3–5. As a result of this study, I will offer
recommendations for improvement to the school to better support teachers in teaching
mathematics in Grades 3–5. You are eligible to participate in this study as you have been
teaching at the school for at least 3 years and are a classroom teacher in Grades 3–5.
Participation in this study is voluntary and teachers will be chosen in the order of
acceptance received for a maximum of six teachers across Grades 3–5. You may choose not to
participate in this study. If you do choose to participate in this study, you will be asked to
participate in one 1-hour interview about mathematics teaching and learning, one 30-minute
interview prior to teaching to discuss the content of the lessons, and two 45-minute classroom
observations within 2 weeks. All data will remain strictly confidential. Thank you in advance for
your participation.

Abstract (if available)

Abstract Ensuring that all children have opportunities to engage in deep mathematical learning is essential for our future. Children must learn to value, appreciate, and use mathematics, and see themselves as knowers and doers of mathematics. Teachers are the key to making this happen and to be successful, many things need to be in place. The purpose of this study was to understand the knowledge, motivation, and organizational influences that support teachers’ ability to enact equitable mathematics teaching practices (EMTPs) that support equitable opportunities for all in mathematics. Literature revealed several assumed knowledge, motivation, and organizational influences related to the enactment of EMTPs. Knowledge factors included knowing and understanding the goals of mathematics and math teaching, knowing and understanding teaching practices that support equity in the mathematics classroom, knowing and understanding content knowledge and progressions, as well as knowing how to enact EMTPs. Teacher efficacy, as well as collective efficacy, were revealed as motivation factors. Identified organizational influences included having a common vision and philosophy, providing ongoing professional learning, and having a standards-based curricular resource aligned to the philosophy. Through an adapted gap analysis based on Clark and Estes's (2008) gap analysis framework, both needs, and assets, were revealed. Six classroom teachers in Grades 3–5, all of whom had been teaching at the school for at least 3 years, took part in this qualitative study. Interviews and observations revealed that teachers have high teacher efficacy for impacting their students’ learning; however, collective efficacy, as well as all knowledge and organization influences, resulted in needs. Recommendations, as well as implementation and evaluation plans, are shared in the final chapters of this dissertation.

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Gaskell, Gynelle DeAun (author)

Core Title The enactment of equitable mathematics teaching practices: an adapted gap analysis

School Rossier School of Education

Degree Doctor of Education

Degree Program Educational Leadership (On Line)

Degree Conferral Date 2022-08

Publication Date 06/13/2022

Defense Date 04/28/2022

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

Tag deep mathematical learning,equitable mathematics,equitable mathematics teaching practices,equitable opportunities for mathematics learning,equity in mathematics,gap analysis,KMO,OAI-PMH Harvest

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Picus, Lawrence (committee chair), Copur-Gencturk, Yasemin (committee member), Robles, Darline (committee member)

Creator Email ggaskell@usc.edu,gynelle@me.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC111339680

Unique identifier UC111339680

Document Type Dissertation

Format application/pdf (imt)

Rights Gaskell, Gynelle DeAun

Internet Media Type application/pdf

Type texts

Source 20220613-usctheses-batch-946 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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Repository Name University of Southern California Digital Library

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