LEVERAGING MATHEMATICAL JUSTIFICATION OPPORTUNITIES TO SUPPORT 
STUDENTS’ AGENCY IN THE LEARNING PROCESS: A CASE STUDY OF ONE 
TEACHER’S PRACTICE AND PERSPECTIVES 

By 

Kristin Giorgio 

A DISSERTATION 

Submitted to 
Michigan State University 
in partial fulfillment of the requirements   
for the degree of 

Curriculum, Instruction, and Teacher Education – Doctor of Philosophy 

2025 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ABSTRACT 

Mathematical reasoning is essential to knowing and doing mathematics, and 

mathematical reasoning practices can be leveraged by teachers to support student voice and 

choice in the learning process. This dissertation explored how one elementary teacher provided 

opportunities for students to engage in the mathematical reasoning practice of justification and 

how these opportunities supported her goals for equity in a classroom of diverse students. Given 

that agency emerged as a prominent theme in the teacher’s equity goals, this dissertation focused 

analysis of findings on the relationship between mathematical justification and agency, with 

connections to interrelated equity constructs (such as mathematical identity) expressed by the 

teacher, as well. Mathematical justification can be defined as, “the process of supporting your 

mathematical claims and choices when solving problems or explaining why your claim or 

answer makes sense” (Bieda & Staples, 2020, p. 103). Case study methodology was used to 

examine the teacher’s curriculum materials, classroom enactment strategies, equity goals, and 

perceptions on the relationship between justification and equity. This study was situated in a self-

contained, racially/ethnically diverse, fourth-grade classroom in an urban district in the Midwest, 

taught by a female Teacher of Color. Findings are reported in the form of three articles, each 

discussing connections between justification and equity (with a focus on agency). Each article 

was written for different primary audiences: the first for mathematics education researchers, the 

second for mathematics teacher educators, and the third for practitioners.  

The first article presents the broadest overview of how and why the teacher provided 

justification opportunities, with the second two articles focusing more in-depth on specific types 

of justification opportunities. This first article, which is Chapter 2 of the dissertation, was written 

for researchers as the primary audience and will be submitted for publication consideration to the 

Journal of Mathematics Teacher Education (JMTE). This article was guided by two research 

questions: (1) How does a teacher facilitate opportunities for students to engage in the 

mathematical practice of justification? (2) How does the teacher perceive mathematical 

justification as supporting her equity goals? Findings showed that the teacher was committed to 

regularly incorporating justification opportunities in her mathematics lessons. She facilitated 

justification opportunities through purposeful task design and enactment strategies, situated 

within a conducive classroom environment. These opportunities were driven by her equity goals, 

which included a focus on supporting student agency in the learning process and cultivating the 

 
development of positive mathematical identity in students. Given how infrequent justification is 

implemented in classrooms, and the limited research on the relationship between justification 

and equity in the literature, these findings help illuminate how justification can be leveraged by a 

teacher in ways relevant to their classroom context and equity goals.  

The second manuscript (Chapter 3) was written for mathematics teacher educators as the 

primary audience and is intended to be submitted for publication consideration to the 

Mathematics Teacher Educator (MTE). This article was guided by two research questions: (1) 

What are one teacher’s goals for promoting equity in the classroom? (2) How does the teacher 

leverage the practice of mathematical justification to meet her equity goals? Findings showed 

that facilitating students’ agency and supporting students in developing positive mathematical 

identity were the primary equity goals for the teacher. Advancing these goals involved engaging 

students in problem-based learning experiences that focused on discovering and justifying 

mathematical patterns and relationships in ways that centered the students’ ideas and reasonings. 

The article elaborates how and why the nature of problem-based learning experiences opened up 

spaces for leveraging justification to meet these equity goals. 

The third manuscript (Chapter 4) is a practitioner-focused piece written for the journal, 

Mathematics Teacher: Learning and Teaching PK-12 (MTLT). This article discusses how the 

beginning of curricular units can be conducive places to facilitate justification opportunities in 

ways that support students’ engagement with new content and agency in the learning process. 

The article describes how and why the teacher in this study facilitated justification opportunities 

at the beginning of units and offers guidance for other teachers hoping to modify their practice to 

provide their students with more mathematical justification opportunities. 

 
 
 
 
 
 
ACKNOWLEDGEMENTS 

I would like to thank my Dissertation Director, Dr. Corey Drake, and my Committee Chair, 

Dr. Tonya Bartell, for your ongoing support and guidance throughout the process of conducting 

this study and writing this dissertation. I am so grateful for the multiple rounds of feedback you 

provided on my writing to help me grow as a scholar. Beyond this study, thank you both for your 

advising, support, and words of encouragement throughout my graduate school journey. I would 

also like to thank my additional committee members, Dr. Kristen Bieda, Dr. Sandra Crespo, and 

Dr. Beth Herbel-Eisenmann, for your roles and support along the way. I am so grateful to have 

had such a large dissertation committee with a vast and diverse set of knowledge and experiences 

to help inform the design of my study and analysis of my work. Thank you also, “Ms. Josie,” for 

welcoming me into your classroom and sharing your time, knowledge, and practice with me. I 

learned so much from you, and your students are so fortunate to have you as their teacher. I would 

like to thank my family and friends who have encouraged me over the years to pursue my passions 

and goals. Thank you for believing in me and supporting me through my graduate school and life 

journey. 

iv 

 
 
TABLE OF CONTENTS 
CHAPTER 1: INTRODUCTION ................................................................................................... 1 
REFERENCES ......................................................................................................................... 14 

CHAPTER 2: EXPLORING HOW AND WHY ONE TEACHER FACILITATED 
MATHEMATICAL JUSTIFICATION OPPORTUNITIES TO SUPPORT HER GOALS FOR 
EQUITY ........................................................................................................................................ 17 
REFERENCES ......................................................................................................................... 47 

CHAPTER 3: THE ROLE OF JUSTIFICATION-ORIENTED PROBLEM-BASED LEARNING 
EXPERIENCES IN SUPPORTING STUDENTS’ AGENCY AND MATHEMATICAL 
IDENTITY .................................................................................................................................... 51 
REFERENCES ......................................................................................................................... 79 

CHAPTER 4: INCORPORATING MATHEMATICAL JUSTIFICATION AT THE 
BEGINNING OF UNITS TO SUPPORT STUDENTS’ ENGAGEMENT WITH NEW 
CONTENT AND AGENCY IN THE LEARNING PROCESS ................................................... 81 
REFERENCES ......................................................................................................................... 94 

CHAPTER 5: CONCLUSION ..................................................................................................... 95 
REFERENCES ....................................................................................................................... 102 

v 

 
 
 
 
 
 
CHAPTER 1: INTRODUCTION 

Teacher practice plays a key role in advancing equitable experiences and outcomes for 

students. Specifically, teacher practice that facilitates mathematical reasoning opportunities can 

advance many aspects of equity. For example, engaging students in mathematical reasoning can 

strengthen students’ problem-solving skills, promote conceptual understanding of content, and 

contribute to productive dispositions towards mathematics (Boaler, 2022; Staples et al., 2012). 

Justification is a particularly powerful reasoning practice because it can be leveraged to engage 

students in the process of making their own mathematical discoveries, defend claims and 

conjectures they make during that discovery process, develop sensemaking to understand why 

their ideas or their peers’ ideas might make sense, and build knowledge together as a classroom 

community. Justification can provide opportunities for students to take ownership of the 

mathematics, in ways that center their voices, perspectives and ideas, sharing in authority of 

knowledge with the teacher (Ball & Bass, 2008; Bieda & Staples, 2020). Bieda and Staples 

(2020) define mathematical justification as, “the process of supporting your mathematical claims 

and choices when solving problems or explaining why your claim or answer makes sense (p. 

103).” Justification can be considered a “learning practice” because it supports students in 

learning and doing many aspects of mathematics (Cohen & Ball, 2001; Staples et al., 2012). This 

practice is further supported in the Standards for Mathematical Practice, particularly the 

practice, construct viable arguments and critique the reasoning of others, which specifies 

students, “justify their conclusions, communicate them to others, and respond to the arguments 

of others” (National Governors Association Center for Best Practices & Council of Chief State 

School Officers, 2010, p. 6-7). Bieda and Staples (2020) describe justification as an “equity 

practice” because it supports students’ access to content and agency in the learning process. With 

regard to agency, Bieda and Staples (2020) state that providing students’ opportunities to justify 

their reasoning builds their agency since students are generating mathematical ideas themselves 

in ways that make sense to them rather than practicing doing someone else’s mathematics. This 

dissertation focuses primarily on the relationship between justification and agency, because 

agency emerged as a prominent equity goal for the teacher in the study. By agency, I refer to 

when students are empowered to have influence in their learning environment, by which they 

play a role in their own mathematical development, and when they learn from/with their peers 

  1 

through the decentering of the teacher as a mathematical authority (Grootenboer & Jorgensen, 

2009).  

The mathematical reasoning practice of justification has benefits for both students’ 

learning and teachers’ instruction. Similarly, as described above, for students, the justification 

process can support students in talking about mathematics and learning from one another’s 

contributions, facilitate their agency in the learning process, increase engagement, strengthen 

reasoning skills, support conceptual understanding of content, and foster positive mathematical 

identities (Bieda & Staples, 2020; Boaler, 2022; Staples et al., 2012). For teachers, eliciting 

students’ justifications can help with assessing students’ working understanding of content at a 

given time (Staples et al., 2012). This can help teachers be responsive to or build from what 

students know in ways that facilitate students’ agency to guide them(selves) to the new content 

to be learned. 

While there are many benefits to incorporating justification into mathematics instruction, 

it is a practice that is lacking in classrooms (Jacobs et al., 2006; Melhuish et al., 2020). One 

reason could be that some curriculum materials do not offer guidance to teachers on how to 

support justification (Staples et al., 2012). However, Bieda et al. (2010) found that even with 

curriculum materials that did provide some guidance, many teachers still did not engage students 

in justification opportunities. One reason for this could be that many teachers have limited 

conceptions and experiences with justification (Knuth, 2002; Staples et al., 2012). For instance, 

many teachers did not experience learning how to develop justifications as mathematics students 

themselves, or justification may not have been a prominent focus in their methods courses during 

teacher preparation. Additionally, standardized tests, which predominantly include multiple-

choice questions to be able to quickly assess many students’ answers to questions at scale, tend 

to focus on procedural fluency over argumentation practices like justification. Since various 

federal or district policies hold teachers accountable to their students’ standardized tests scores in 

some capacity (NCLB Act of 2001, 2002), it makes sense that teachers’ instruction would 

continue to be influenced by a focus on procedures with less attention to reasoning practices like 

justification. While policies can influence teacher practice, so can a teacher’s goals and beliefs 

(Aguirre & Speer, 1999). Given the various reasons why justification may not be a prominent 

practice despite its many benefits, this dissertation sought to examine why one teacher perceived 

justification to be an important practice for advancing her equity goals and how she facilitated 

  2 

justification opportunities to meet those goals. Furthermore, I argue that understanding how the 

justification process is facilitated can be better understood when analyzed through the lens or 

context of the teacher’s equity goals. Given there is limited research in the field on justification 

as a process (Karunakaran & Levin, 2022) and a large gap in the literature on research that 

examines the relationship between justification and equity (Bieda et al., 2023), this dissertation 

sought to better understand how justification opportunities can be leveraged to meet a teacher’s 

equity goals.    

Conceptualizing Equity 

There are many conceptualizations of the meaning of equity in the field (Aguirre et al., 

2013; Cochran-Smith & Keefe, 2022; Gutiérrez, 2009; Horn, 2012). Cochran-Smith and Keefe 

(2022) conducted a literature review on the word “equity” and identified three themes in the 

discourse on equity in teacher education literature: (1) Equity as a by-product of accountability, 

(2) Equity as differentiated from equality, and (3) Equity as a result of “best practice.” 

Regarding the meaning of the first, this refers to the many policies and initiatives that were 

designed to make teacher education more accountable to the public, such as rigorous teacher 

licensure, accreditation standards, and teacher evaluation measures. The assumption was that 

high teacher quality results in high student achievement, suggesting that all students succeeding 

or achieving at higher levels means more equitable student outcomes. The second theme refers to 

the notion that equity does not mean equal, fair, or same, which recognizes that students are 

unique with different learning styles, circumstances, and needs and can thus require different 

resources to learn the same content. Teacher education studies explain that learning differences 

among students are not a result of deficits in students, but rather, all students have strengths and 

bring cultural funds of knowledge which teachers can be responsive to and leverage to support 

their access to content and thus achieve equitable outcomes. The third theme centers around 

equity being a result of “best practice,” which says that there are “high-leverage practices” or 

“core practices” (e.g., facilitating discussion, eliciting and interpreting student thinking, 

coordinating and adjusting instruction) that teachers can do to promote equitable experiences and 

outcomes for students (Ball & Forzani, 2009). This dissertation study focused on equity as 

related to the third theme, given the focus on understanding mathematical justification as a key 

practice for advancing equity for students. 

  3 

Considering the vast use of the word equity in the literature, Cochran-Smith and Keefe 

(2022) distinguish between what they call thin equity and strong equity. In defining thin equity, 

Cochran-Smith and Keefe (2022) describe, “When we talk about thin equity in teacher education 

policy and practice… we refer to initiatives that assume equity has to do with an individual’s 

access to education resources” (p. 19; emphasis in original). In describing strong equity, 

Cochran-Smith and Keefe (2022) explain, “We refer to strong equity to acknowledge the 

complex and intersecting historical, economic, and social systems that create inequalities in 

access to teacher quality in the first place” (p. 19, emphasis in original). Essentially, thin equity 

describes policies and practices that center around individualism and neoliberalism, whereas 

strong equity describes the complexities of learning as being socially, historically, and 

economically interrelated. A focus on strong equity can lead to positive social change and better 

learning experiences and outcomes for students, particularly students from historically and 

currently marginalized groups, such as Black and Latinx students.  

Gutiérrez (2009) also notes different uses of equity in the field, specifically within 

mathematics education. Gutiérrez (2009) explains there has been a “dominant” discourse on 

equity that has centered around access to mathematics content and achievement in the subject. 

Access refers to resources and opportunities (e.g., high quality teachers, curriculum materials, 

tools) that allow each individual student to fully participate in mathematics, and achievement 

refers to student outcomes on institutionally acknowledged performance and participation 

measures (e.g., standardized tests). These dominant dimensions of equity have similar 

characterizations to the individual and neoliberal notions of thin equity described by Cochran-

Smith and Keefe (2022). Gutiérrez (2009) states that to attain “true equity” (p. 6) for students, 

the field must also center constructs of power and identity, which are described as critical 

dimensions of equity, to also support students’ access to and achievement in mathematics. 

Gutiérrez describes the identity dimension of equity as students seeing themselves reflected in 

and as doers of mathematics. Power is characterized by student empowerment, voice, and 

influence in the mathematics classroom and beyond. Gutiérrez’s frames these four equity 

dimensions by situating them on intersecting axes to illustrate that they are interrelated and work 

together to fully realize equity. Access and achievement comprise the “dominant axis,” and 

identity and power comprise the “critical axis.” Gutiérrez (2009) calls for an increased focus on 

the identity and power dimensions, given the dominant focus in the field (historically and 

  4 

currently) on the constructs of access and achievement. I argue that mathematical justification, as 

driven by a teacher’s disposition that includes a focus on critical dimensions of equity, can be a 

powerful practice for advancing equity for students, particularly in a racially/ethnically diverse 

classroom of students. 

I, as supported by others in the literature, view agency to be a critical equity construct. 

Bieda and Staples (2020) and Berry (2020) both view agency as an equity construct, interrelated 

with other equity dimensions. Specifically, Bieda and Staples (2020) view agency and access as 

interrelated equity constructs, and Berry (2020) views agency and identity as interrelated equity 

constructs. Bieda and Staples (2020) define agency as “when students develop a sense that they 

can do mathematics and generate mathematical ideas; each student sees mathematics as a tool to 

use,” and access as referring to, “each student in a class having opportunities to engage in 

rigorous mathematics and to learn mathematics that is meaningful” (p. 104). Bieda and Staples 

(2020) describe agency and access as “intertwined,” stating that students’ agency is developed in 

relation to the opportunities they are provided (access), and students’ access to mathematics is 

also shaped by how they engage with content and challenges, participate in class, and perceive 

themselves as capable (agency). Berry (2020) uses NCTM’s (2014) Principles to Actions 

framing of equity to describe how agency and identity are interrelated constructs that can 

advance equity. In defining agency, Berry (2020) cites Aguirre et al. (2013) and situates identity 

in the definition of agency, “Agency is our identity in action and the presentation of our identity 

to the world” (slide 15). In other words, when a student identifies as being good at mathematics, 

they adopt behaviors and actions associated with smartness in mathematics; and when students 

have the opportunities to make choices and decisions associated with agency in the learning 

process, this is cultivating and reflecting their mathematical identities. Chapter 3 provides further 

elaboration of Berry’s (2020) framing of agency and identity as interrelated constructs that can 

advance equity.  

Given that agency and identity are interrelated equity constructs (Berry, 2020) and 

identity is part of the critical axis (Gutiérrez, 2009), I view agency to be an equity construct, 

situated on the critical axis. The construct of agency takes into account the learning environment 

as a social (not individual) system, where students feel empowered to use their voice, make their 

own decisions, have their ideas centered, and learn from one another, sharing in authority of 

knowledge with the teacher. Having agency provides opportunities for students’ identities to be 

  5 

 
reflected in the learning process, and in turn, they are building positive mathematical identities 

by seeing themselves as capable knowers and doers of mathematics. Because having agency 

means a more active and student-centered learning process (e.g., versus passively receiving 

information via lecture to be memorized and regurgitated, whereby conceptual understanding 

becomes less certain), this can result in more active student engagement, increased access to 

content, deeper understanding of mathematics, and thus higher achievement outcomes. Webb et 

al. (2014) found that teacher practice that supported student agency through centering their ideas 

and reasonings led to increased participation, engagement, and achievement in mathematics. For 

these reasons, I view agency as a critical equity construct. Chapter 3 elaborates the relationship 

between agency and identity (with connections to justification), from the literature and from the 

perspective of the teacher participant in the study.  

Bieda and Staples (2020) define justification as an “equity practice” that can advance 

agency. Justifying one’s reasoning supports agency because students are tasked with generating 

and defending mathematical ideas themselves in ways that make sense to them, rather than 

practicing doing the teacher’s mathematics. Bieda, Staples, and colleagues (2023) situate their 

conceptualization of justification from their earlier work (i.e., Bieda & Staples, 2020) between 

identity and access on Gutiérrez’s axes. They view identity as part of agency, and access and 

agency as intertwined equity constructs that can be advanced through justification. Figure 1 

shows the image they created in Bieda et al. (2023) to illustrate where they situate their 

conceptualization of justification on Gutiérrez’s axes.  

Figure 1. Bieda et al.’s (2023) Conceptualization of Justification (Bieda & Staples, 2020) on 
Gutiérrez’s Dimensions of Equity Axes 

Gutiérrez (2009) states that all four dimensions are necessary for attaining “true equity” 

(p. 6) and explains that the critical axis and the dominant axis have an inverse relationship which 

  6 

 
 
can support one another. “Learning dominant mathematics may be necessary for students to be 

able to critically analyze the world, while being able to critically analyze the world may provide 

entrance into dominant mathematics” (p. 6). Thus, a focus on identity and power is essential for 

supporting students’ access to and achievement in mathematics. I argue that justification 

opportunities (which can center students’ ideas, perspectives, claims, and decisions) can advance 

all aspects of equity, as supporting students’ identity and power in mathematics can, in turn, 

promote access and achievement. Given Gutiérrez’s (2009) claim that identity and power do not 

frequent the dominant discourse of equity in mathematics (yet are necessary for advancing 

equity), I argue that a focus on the critical dimensions will be important for justification 

opportunities to truly advance equity for students, particularly in a classroom of 

racially/ethnically diverse students.  

Leveraging justification to support critical aspects of equity, such as agency, requires the 

teacher to share authority of knowledge with students. I draw on Zavala and Aguirre’s (2024) 

conceptualization of distributing intellectual authority to frame how justification opportunities 

can be leveraged to support critical dimensions of equity. Zavala and Aguirre (2024) define 

distributing intellectual authority as involving, “positioning students as mathematical authorities 

in the classroom, using participation structures that expand student mathematical contribution 

opportunities, and embracing the role of teacher as facilitator and co-constructor of knowledge in 

the classroom” (p. 99). This definition encompasses the constructs of agency, identity, and 

justification, as it mentions providing students the autonomy to have their ideas and identities 

reflected in the learning process. Zavala and Aguirre (2024) describe three “instructional 

approaches” involved in distributing intellectual authority: (1) position students as holders of 

math knowledge, (2) expand use of collaborative participation structures, and (3) embrace the 

teacher-as-facilitator role. Zavala and Aguirre’s (2024) conceptualization of distributing 

intellectual authority is a helpful framework for explaining ways in which justification 

opportunities can support critical dimensions of equity, such as advancing students’ agency and 

cultivating positive mathematical identities, which will be elaborated in Chapter 3. Figure 2 

shows where I situate justification (when leveraged with an intentional focus on agency) on 

Gutiérrez’s axes. Chapter 3 provides the most in-depth elaboration of the relationship among 

justification, agency, and identity. 

  7 

 
Figure 2. Conceptualization of Justification on Gutiérrez’s Dimensions of Equity Axes 

In this dissertation, I conceptualize equity as strong equity (Cochran-Smith & Keefe, 

2022) and considered all four dimensions of equity (Gutiérrez, 2009) in the selection of the 

teacher participant and analysis of her equity goals. Because my views on equity align with 

strong equity (given its focus on learning being influenced by multiple social systems), and 

because I sought to understand the relationship between critical dimensions of equity and 

mathematical justification, the participant I selected was a teacher who I knew also held strong 

equity views, based on the type of school she worked at and my experiences with her as a former 

colleague. More about the depth of her equity goals that were uncovered through the study will 

be elaborated throughout the manuscripts of the dissertation.  

Methodology of the Dissertation 

Research Focus. The research questions guiding this dissertation centered around the 

relationship between justification and equity, with analysis focusing on the teacher’s specific 

equity goals (in which agency emerged as a central theme). The dissertation explored questions 

pertaining to the justification opportunities the teacher provided and her perceptions of the 

impact of these opportunities on students. The overarching research questions guiding the study 

were (1) How does a teacher facilitate opportunities for students to engage in the mathematical 

practice of justification? (2) How does the teacher perceive mathematical justification as 

supporting her equity goals? The specific research questions that guided each manuscript will be 

outlined in the next section, which summarizes each article.  

Case Study Methods. This dissertation was conducted using case study methodology, which 

refers to a “case” bounding the parameters of the study (Merriam & Tisdell, 2015). The case that 

bounded this study was one teacher’s practice and perspectives, regarding justification 

opportunities and her equity goals. The teacher’s practices and perspectives were examined 

  8 

 
 
 
through interviews, classroom observations, and implemented curriculum materials. The study is 

situated in Merriam’s (1998) constructivist pragmatic epistemological assumptions of case study, 

which describes using findings from research to inform practice. One goal of this dissertation 

was to provide insight into how one teacher incorporated justification opportunities to meet her 

equity goals, with the overarching aim to support other teachers in leveraging justification within 

their own contexts and to meet their equity goals. 

School Context and Participant Background. The context of this study was one fourth-grade 

classroom in an urban district in the Midwest, with students predominantly from marginalized 

groups (including Black and Latinx students). The classroom was situated in a school with Title I 

status. Because the majority of students qualified for free or reduced lunch, the school provided 

all students free breakfast, lunch, and snacks during the school day. The school, Midwest 

Academy (pseudonym chosen by the participant), was known for its commitment to place-based 

education and social justice. The teacher in the study defined place-based learning in the 

following way, and relative to the context of her school: 

[Midwest Academy] defines it as creating critical thinkers who contribute to the wellbeing of 

their community. I guess that place based is like putting the learning in the context of your place. 

Sometimes you can solve problems in that place that you know can come from your learning, or 

sometimes it's just you can learn about your place to develop and foster love of it and 

stewardship of it. 

At the grain size of a curricular unit, the teacher described place-based learning as, 

Usually, a place-based unit will begin with a question, I guess, that you're trying to answer. It 

could be about the environment, it could be about history, it could be about perception, it could 

be about identities. You're using your place to answer, I think they call it, an essential question. 

The participant in this study was the teacher of this fourth-grade classroom. The teacher, 

Josie (her chosen pseudonym) was selected from a combination of convenience sampling 

methodology due to proximity and a pre-existing relationship I had with the teacher and 

purposeful sampling methodology based on practices and dispositions held by the teacher as 

someone hired to teach at a place-based school (Merriam & Tisdell, 2015). From having worked 

together at the same school for one year, I recalled that Josie was passionate about creating 

community-oriented and culturally relevant lessons, and that all teachers at Midwest Academy 

received yearly professional development on place-based learning. Place-based learning is 

situated in strong equity perspectives, given how place-based learning recognizes socio-cultural 

  9 

and community-based knowledge. I knew incorporating student voice and ideas into 

mathematics instruction was important to Josie, and I was curious to learn more about how she 

conceptualized the relationship between justification and equity. Josie identifies as a Filipina 

woman and immigrated to the United States at age 13, which could have influence on her 

commitment to connecting culture, community, and school content because experiencing 

schooling in two countries informs her perspectives on education, and I know staying connected 

to her culture is important to her. At the time of the study, Josie had nine years of teaching 

experience, with six of those years being at Midwest Academy.  

Researcher Positionality. Given that I taught at Midwest Academy at one point when Josie was 

a teacher there as well, I brought a somewhat insider lens to the research context. For example, I 

knew that Josie was passionate about place-based education, social justice, and student-centered 

learning. Given the potential for mathematical justification to advance critical aspects of equity, I 

was curious about how Josie thought about this connection and how she facilitated student-

centered justification opportunities in her classroom. Since I never collaborated with Josie in 

lesson planning or observed her instruction when I worked at the school, I did not have a full 

understanding of her practice and perspectives and thus had more of an outsider lens with regard 

to her own classroom. This insider/outsider perspective I brought suggested to me that Josie’s 

practice and perspectives could provide an interesting context to explore how justification 

opportunities are facilitated in ways that might relate to or advance critical dimensions of equity, 

given that these dimensions are left out of dominant discourse but crucial for realizing, what 

Gutiérrez (2009) calls, “true equity.”  

It is important to acknowledge that I brought to this research a particular status, as a 

White researcher affiliated with a prominent university who entered a classroom space of 

predominantly Students of Color and a Teacher of Color. Because I knew my presence could 

have some influence on the space, it was important to me that I learned from a teacher with 

whom I had an established and trusting relationship with and who remembered me as a teacher at 

one point. Josie knew that my role was to learn from her. Mindful of this status I brought, I tried 

to decenter myself by quietly observing at the edge of the classroom, and by centering Josie’s 

voice during our meetings by asking questions that positioned her as expert and never offering or 

answering questions about my opinions. The insider/outsider lens I brought allowed for the 

  10 

selection of, and deep learning from, an exceptionally equity-minded teacher and a trusting 

researcher-participant relationship. 

Summary of Manuscripts 

This dissertation contains three manuscripts, each intended to be submitted to a different 

journal. All have an overarching focus on mathematical justification. Given that supporting 

student agency emerged as a prominent equity goal for the teacher in the study, all three 

manuscripts discuss connections between justification and agency. Each manuscript also 

discusses other interrelated constructs within Josie’s equity goals. For example, Manuscripts One 

and Two elaborate Josie’s connections to mathematical identity. The intention of writing three 

articles with the same overarching focus (on justification and agency) but each for a different 

primary audience (researchers, teacher educators, and practitioners, respectively), is to support 

the ongoing challenge in the field of bridging research and practice, aligning with constructivist 

pragmatic case study methodology.  

Manuscript 1: Researcher-focused Audience. The intended journal for the submission of this 

manuscript was the Journal of Mathematics Teacher Education (JMTE). The primary audience of 

JMTE is teacher education researchers. Two research questions guided this paper: (1) How does 

a teacher facilitate opportunities for students to engage in the mathematical practice of 

justification? (2) How does the teacher perceive mathematical justification as supporting her 

equity goals?  

Findings indicated that Josie was committed to regularly implementing justification 

opportunities in her mathematics lessons. She did this through purposeful task design and 

enactment strategies, and by establishing a conducive classroom environment. These 

opportunities were all driven by her personal equity goals, which focused on increasing student 

agency in the learning process. Given the intertwinement of agency and identity, Josie’s 

perspectives on the relationship between agency, identity, and justification are also elaborated. 

This study helps fill gaps in the literature that suggest a limited understanding on the relationship 

between mathematical justification and critical dimensions of equity (Bieda et al., 2023). A 

better understanding of mathematical practices (such as justification) that can advance students’ 

agency and positive mathematical identity development is important for supporting equitable 

outcomes for students in mathematics. 

  11 

 
Manuscript 2: Teacher Educator-focused Audience. This manuscript was written with the 

intention to submit to the Mathematics Teacher Educator (MTE). The primary audience of MTE 

is teacher educators. This paper was guided by the following two research questions: (1) What 

are one teacher’s goals for promoting equity in the classroom? (2) How does the teacher leverage 

the practice of mathematical justification to meet her equity goals? 

For Josie, advancing equity in mathematics meant a focus on facilitating students’ agency 

and supporting students in building positive mathematical identity, which she saw as intertwined. 

This involved engaging students in problem-based learning experiences that centered on noticing 

and justifying patterns, rather than imparting these herself. This meant students were generating 

and justifying their own mathematical concepts, rather than solely receiving and practicing 

Josie’s mathematics. When examining the justification opportunities that Josie provided, 

problem-based learning emerged as a prominent theme. This article discusses Josie’s 

perspectives on student agency and identity in mathematics. The paper presents two cases of how 

she facilitated justification opportunities through problem-based learning experiences to support 

her equity goals. These cases are important for mathematics teacher educators because they 

provide examples, as described within one classroom context, of how and why problem-based 

learning can open spaces for leveraging justification opportunities to advance equity for students.  

Manuscript 3: Practitioner-focused Audience. This manuscript will be submitted to the 

journal, Mathematics Teacher: Learning and Teaching PK-12 (MTLT). MTLT is a practitioner 

journal, with in-service and pre-service K12 teachers as the primary audiences. This article 

discusses how to incorporate opportunities for mathematical justification at the beginning of 

units to support students’ engagement with new content and their development of agency in the 

learning process. The beginning of Josie’s fractions unit is provided as an example. Shared are 

the curricular tasks she implemented at the start of the unit, excerpts of dialogue from classroom 

discussion, Josie’s thoughts after debriefing the lesson, and guidance for other teachers. 

The three manuscripts in this dissertation illustrate a powerful connection between 

justification and agency (with some attention to other interrelated equity dimensions) from the 

perspective of one Teacher of Color and her students in a racially/ethnically diverse classroom of 

students. Findings are written in the form of three articles, intended for publication consideration 

in three journals, with the goal of reaching three major audiences in the field of education: 

researchers, teacher educators, and practitioners. The next chapter, Chapter 2, presents the 

  12 

broadest overview of how and why Josie provided justification opportunities, (with Chapters 3 

and 4 focusing more in-depth on specific types of justification opportunities).  

  13 

 
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mathematics learning and teaching: Rethinking equity-based practices. National Council 
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teacher practice. The journal of mathematical behavior, 18(3), 327-356. 
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education. Journal of teacher education, 60(5), 497-511. 
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Bieda, K., Staples, M., Doherty, K., Matthews-Johnson, A., & Hwang, J. (2023). Considering the  

alignment between teachers’ conceptions of justification and their visions for equitable 
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Education, vol. 1, 124-130. Reno, NV. 

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Learning and Teaching PreK-12, 113(2), 102–108. 
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opportunities. Journal for Research in Mathematics Education, 41(4), 351–382. 
https://doi.org/10.5951/jresematheduc.41.4.0351 

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Gutiérrez, R. (2009). Framing equity: Helping students “play the game” and “change the game.”   

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Teaching for Excellence and Equity in Mathematics, 1(1), 4-8. 

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classrooms. Reston, VA: National Council of Teachers of Mathematics.   

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eighth-grade mathematics teaching in the United States align with the NCTM standards? 
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Karunakaran, S. S., & Levin, M. (2022). Justification in the Context of Tertiary Mathematics:  

Undergraduate Students Exploring the Properties and Relations of the Dihedral Group. In 
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(pp. 239-250). Springer, Cham. 

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Research in Mathematics Education, 33(5), 379–405. https://doi.org/10.2307/4149959 

Melhuish, K., Thanheiser, E., & Guyot, L. (2020). Elementary school teachers’ noticing of  
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4 

Merriam, S. B. (1998). Qualitative Research and Case Study Applications in Education. Revised  

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implementation. John Wiley & Sons. 

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mathematical success for all. Reston, VA. 

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Officers. (2010). Common core state standards for mathematical practice. 
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No Child Left Behind (NCLB) Act of 2001, 2002: Qualifications for Teachers and Professionals. 

Responsive Classroom. (2020). Responsive classroom: Elementary core course resource book. 

Staples, M. E., Bartlo, J., & Thanheiser, E. (2012). Justification as a teaching and learning  

practice: Its (potential) multifaceted role in middle grades mathematics classrooms. The 
Journal of Mathematical Behavior, 31(4), 447-462. 
https://doi.org/10.1016/j.jmathb.2012.07.001  

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  15 

 
 
 
 
 
 
 
(2014). Engaging with others’ mathematical ideas: Interrelationships among student 
participation, teachers’ instructional practices, and learning. International Journal of 
Educational Research, 63, 79–93. https://doi.org/10.1016/j.ijer.2013.02.001 

Zavala, M. R., & Aguirre, J. M. (2024). Cultivating mathematical hearts: Culturally responsive  

mathematics teaching in elementary classrooms. Corwin Press. 

  16 

CHAPTER 2: EXPLORING HOW AND WHY ONE TEACHER FACILITATED 
MATHEMATICAL JUSTIFICATION OPPORTUNITIES TO SUPPORT HER GOALS 
FOR EQUITY 

Abstract  

Teacher practice plays an important role in advancing equitable experiences and 

outcomes for students. Teacher practice that encourages students to engage in mathematical 

reasoning that promotes inquiry with justification can support students’ agency and engagement 

with content in the learning process. This study examined how one fourth-grade Teacher of 

Color leveraged the mathematical practice of justification to advance her goals for equity in a 

classroom of racially/ethnically diverse students. Findings indicated that the teacher was 

committed to regularly implementing justification opportunities in her mathematics lessons. She 

did this through purposeful task design and enactment strategies, and by establishing a conducive 

classroom environment. These justification opportunities were driven by her personal equity 

goals, which were focused on advancing students’ agency in the learning process and cultivating 

students’ development of positive mathematical identities.  

Background and Objectives 

Mathematical reasoning has been described as an inherent component of what it means to 

know and do mathematics (Ball & Bass, 2003). Scholars have examined the role that 

mathematical reasoning practices can serve in advancing equitable experiences and outcomes for 

students (Bartell et al., 2017; Boaler & Staples, 2008). Inquiry-oriented lessons that leverage 

mathematical reasoning practices can improve students’ problem-solving skills, promote 

conceptual understanding of content, and contribute to productive dispositions towards 

mathematics (Boaler, 2022). Discussion-based classrooms that center students’ mathematical 

reasonings as a basis for forming collective understanding also serve to build and uphold the 

skills and dispositions of democracy (Ball & Bass, 2008). Justification is a particularly powerful 

reasoning practice because it can be leveraged to support students in making their own 

mathematical discoveries, defend claims and conjectures they make during that discovery 

process, and build mathematical knowledge together through sensemaking as a classroom 

community. Justification opportunities can support students in taking ownership of the 

mathematics, as it opens spaces for teachers to center their voices, perspectives, and ideas rather 

than regurgitating and practicing the teacher’s mathematics (Ball & Bass, 2008; Bieda & Staples, 

2020). 

  17 

Mathematical reasoning is characterized by providing both explanations and 

justifications for mathematical claims and choices that are made (Cioe et al., 2015). While 

students are used to providing explanations that describe how they solved a problem or what 

mathematical pattern/relationship they notice (i.e., explanation), they typically will not offer a 

justification for why their choice of solution strategy or mathematical pattern/relationship they 

notice makes sense (i.e., justification) unless further prompted (Cioe et al., 2015). Students’ 

predisposition to offer explanations without justifications can be influenced by their past 

experiences learning mathematics, which may have traditionally involved responding to prompts 

from the teacher that elicited explanation of what they did to solve a problem (i.e., “the how”) 

without being pressed enough for the rationale behind it (i.e., “the why”). Cioe and colleagues 

(2015) distinguish between explanation and justification to explain how engaging students in 

mathematical reasoning (an essential component of doing and learning mathematics) involves 

supporting students in moving from “the how” to “the why.” Cioe et al. (2015) defines an 

explanation as articulating the methods for solving a problem (without communicating why the 

method is correct or appropriate) or sharing evidence that some mathematical pattern or 

relationship holds (without communicating why the relationship must hold). In contrast, a 

justification provides reasoning for why a mathematical relationship holds or why a solution 

strategy is appropriate or correctly applied. Essentially, supporting students in moving from 

explaining to justifying involves eliciting rationales from students. Eliciting rationales supports 

students’ full and authentic engagement in mathematical reasoning, because mathematical 

reasoning is characterized by both explanations and justifications.  

Given how essential justification is to the mathematical reasoning process, Staples et al. 

(2012) describes justification as a “core mathematics practice” (Staples et al., 2012, p. 447). 

More specifically, Staples and colleagues (2012) describe justification as a “teaching and 

learning practice,” (p. 447) which can support both teachers’ instruction and students’ learning. 

For example, justification is a practice that teachers can leverage to engage students with the 

content (e.g., in conceptually oriented ways) and with one another (e.g., supporting collaboration 

and collective knowledge building), and it can be used as a tool to assess what students know. 

For students, engaging in justification can support them in learning content in ways that are 

conceptually oriented and argumentation-based, building valued mathematical skills and 

dispositions. Bieda and Staples (2020) describe justification as an “equity practice” (p. 102), 

  18 

explaining that justification is a practice that can be leveraged to support students' access to 

mathematics content and agency in the learning process. 

Given the value that engaging in reasoning practices can provide for students’ learning of 

mathematics, yet the ongoing challenge of bridging the theory of justification into practice in 

mathematics classrooms (Melhuish et al., 2020), and the limited research in the field that 

examines the relationship between justification and equity specifically (Bieda et al., 2023), this 

study sought to better understand how justification opportunities can be leveraged to meet a 

teacher’s equity goals. It would make sense that a teacher who views justification as a practice 

than can be leveraged to meet their equity goals would be more likely to implement it in their 

classroom. Understanding how and why a teacher facilitates justification through the lens of their 

context and equity goals could provide insight into how other teachers might be supported in 

incorporating the practice in ways they view as relevant to their contexts and goals. This study 

examined two questions (1) How does a teacher facilitate opportunities for students to engage in 

the mathematical practice of justification? (2) How does the teacher perceive mathematical 

justification as supporting her equity goals?  

Conceptual Framework 

This study examined how one teacher facilitated opportunities for students to engage in 

the practice of mathematical justification, as defined by Bieda and Staples (2020), to understand 

connections between justification and equity goals, given its value as a student-centered 

reasoning practice, yet limited understanding of its relationship to equity in the existing 

literature. Bieda and Staples (2020) define justification as, “the process of supporting your 

mathematical claims and choices when solving problems or explaining why your claim or 

answer makes sense” (p. 103). Staples et al. (2012) is one study that provides a helpful 

understanding of the relationship between mathematical justification and a teacher’s goals. 

Staples et al. (2012) explored how twelve middle school teachers conceptualized justification 

and why they found it to be a beneficial practice for their students. There were six “purposes” for 

using justification that surfaced with the teachers. The six purposes were: promoting conceptual 

understanding, fostering valued math skills and dispositions, assessing [what students know], 

fostering valued life-long skills and dispositions (e.g., taking a stand and supporting it with 

evidence), managing diversity (e.g., promoting access for a wider range of students), and 

influencing social relationships (e.g., shifting authority of knowledge from the teacher to be 

  19 

shared with the students). While these purposes can be situated within various equity constructs 

(e.g., influencing social relationships relates to agency), this study builds on the work of Staples 

and colleagues (2012) to more specifically elaborate how justification as a practice might be 

leveraged to support a teacher’s goals for equity more broadly.  

Melhuish and colleagues (2020) also situate justification as a teaching and learning 

practice, similar to Staples and colleagues’ (2012) conceptualization, explaining that justification 

is a practice teachers can leverage to support students in doing mathematics and enhance their 

understanding of mathematics. Melhuish et al.’s (2020) definition of justifying integrates 

mathematical goals for using the practice and the role of the classroom community. Melhuish 

and colleagues (2020, p. 37) define justifying as,  

Reason[ing] with meaning of ideas, definitions, mathematical properties, established 

generalizations to   

•  Show why an idea/solution is true; 

•  Refute the validity of an idea; 

•  Give mathematical defense of an idea that was challenged.  

Melhuish and colleagues (2020) elaborate to explain that their definition of justifying 

closely aligns with Stylianides’ (2007, p. 291) definition of proof, 

• 

It uses statements accepted by the classroom community (set of accepted statements) that are 

true and available without further justification; 

• 

It employs forms of reasoning (modes of argumentation) that are valid and known to, or 

within the conceptual reach of, the classroom community; and  

• 

It is communicated with forms of expression (modes of argument representation) that are 

appropriate known to, or within the conceptual reach of, the classroom community.  

Melhuish et al. (2020) explain that the defining feature of justifying is the process of 

building on claims, meanings, and arguments that students accept as true, in order to build 

collective understanding of mathematics as a classroom community. Stylianides’ (2007) 

definition encompasses this with a focus on the justification product (i.e., statement of the 

argument), whereas Melhuish and colleagues (2020) shift their focus to the process or the “act of 

justifying” (p. 38). This shift in focus supported Melhuish and colleagues (2020) in examining 

teachers’ understanding, noticing, and use of the practice. After a two-year professional 

development in 25 elementary schools (with all third- through fifth-grade teachers as 

participants), Melhuish et al. (2020) found that few teachers attended to justification. Melhuish 

  20 

and colleagues (2020) attributed this to teachers holding varying, nonstandard conceptualizations 

of reasoning (which were deeply rooted, and for many teachers, did not change even after being 

exposed to more precise or authentic definitions in professional development), and these 

differences in conceptions affected teachers’ noticings and implementations of justification. 

Given the complexity of justification, it would make sense that teachers can face challenges in 

learning, noticing, and implementing the practice.  

Ball and Bass (2003) found that facilitating justification, (or what they call “the reasoning 

of justification”), is complex work that involves many factors. Facilitating justification 

opportunities specifically with the goal to cultivate skills and dispositions characteristic of a 

democracy in a classroom, involves purposeful task design and pedagogy that supports student 

inquiry and opportunities to learn from one another in ways that build collective understanding 

of mathematics as a classroom community. They define the reasoning of justification as resting 

on two foundations, and describe what is required in building these foundations, 

We characterized two foundations of the mathematical reasoning of justification: a base of public 

knowledge and mathematical language. The reasoning of justification requires pedagogy, and 

design, just as the teaching of any other aspect of mathematics. Students must learn to use 

publicly-established ideas, methods, and language to make, inspect, validate, improve, and 

extend the mathematical knowledge of the class. And teachers must provide the resources and 

create an environment that encourages and makes possible complex student work (p. 43).  

Both Ball and Bass (2003) and Melhuish et al. (2020) describe justification facilitation as 

involving centering students’ ideas and reasonings to build collective understanding of 

mathematical concepts as a classroom community. Centering students is an essential component 

of both justification and many equity frameworks, including the equity frameworks presented in 

Aguirre et al. (2013), Cochran-Smith and Keefe (2022), and Seda and Brown (2021). These 

frameworks describe centering students as eliciting and building from their ideas, perspectives, 

and funds of knowledge rather than regurgitating and practicing the teacher’s mathematics. Ball 

and Bass (2003) explain that facilitating the reasoning of justification is complex work given the 

attention to multiple factors (e.g., task design, pedagogy, context). Similarly, Remillard and 

Heck (2014) describe the curriculum enactment process as also involving the interplay of various 

factors related to task design, pedagogy, values, and context. Remillard and Heck’s (2014) 

framework for conceptualizing the mathematics curriculum enactment process further illustrates 

this complexity, explaining that tasks (and practices) become enacted as influenced by the 

  21 

following factors: curricular (or task) design, district policy around curriculum use, teachers 

(including their pedagogy, beliefs, and goals) and students (including their identities, knowledge, 

and beliefs). Remillard and Heck describe how these are multiple systems that are interacting in 

complex ways to influence classroom activity, which suggests that understanding the enactment 

of any practice (e.g., mathematical justification) involves examining multiple systems. Looking 

just at the design of the justification task or just the teacher’s pedagogy, for instance, may not 

provide insight into the broader picture of how and why justification opportunities get 

implemented in a classroom.  

In Figure 3, I conceptualize justification opportunities as influenced by multiple 

interacting systems situated within one another. The framework starts at the level of the 

curricular task. The impact of the implementation of a written curricular task selected by a 

teacher for a lesson is situated within purposeful enactment strategies (e.g., pedagogical 

strategies such as questioning techniques or discourse moves that promote student inquiry and 

elicit reasoning), and situated within a particular context (e.g., a conducive classroom 

environment), and also situated within a teacher’s equity goals. 

Figure 3. Framework for Conceptualizing a Teacher’s Implementation of Justification 
Opportunities  

A written task that is intended to elicit justification might include key words such as 

prove, conjecture, justify, verify, show, convince (Bieda et al., 2014). These verbs become 

  22 

 
 
situated within tasks of high-cognitive demand, which Stein and Smith (1998) describe as tasks 

that are conceptually (versus procedurally) oriented, multi-step, and open-ended (e.g., does not 

prescribe the use of a particular solution strategy). Considering this definition, a task might say, 

“Calculate the area of the irregular shape. Justify your solution strategy and answer.” Another 

example might be, “Jordan claims that if you double the perimeter of a rectangle, then the area of 

the new rectangle will be double the area of the original rectangle. Prove whether Jordan’s claim 

is true or not.” As mentioned above, Remillard and Heck (2014) explain that a given task can be 

enacted in various ways as influenced by various external factors, including a teacher’s 

pedagogical moves. How a teacher enacts questioning strategies and discourse moves can be 

helpful in supporting students in communicating their thinking about a problem, orienting their 

ideas to one another, and ultimately justifying their ideas. Some examples of discourse moves, 

adapted from Chapin et al. (2009) and Cioe et al. (2015), that could support students during the 

mathematical justification process include, “Why would it make sense that…?”  “Who agrees or 

disagrees with X, and why?” and “How could you convince someone that doesn’t believe you?” 

It is important for curricular tasks and their enactment to be situated within a conducive 

classroom environment, which is carefully cultivated by the teacher. This includes a classroom 

culture where students feel safe sharing their ideas and taking risks (Klosterman, 2016). Because 

being good at mathematics has traditionally been associated with quickness and correctness, 

producing high levels of math anxiety across students in mathematics classrooms, norms that 

value more than this as constituting mathematical “smartness” (Boaler, 2015; Cohen et al., 1999) 

could support students in feeling more comfortable talking through their in-progress or “rough 

draft math” ideas (Jansen, 2020) during the justification process. Norms shape the type of 

mathematical activity that occurs in a classroom (Cobb et al., 1992). They shape the 

development of mathematical routines and practices, and they shape interaction among students 

and between teacher and students (Yackel & Cobb, 1996). As a result, norms influence what 

counts as valued mathematical contributions (Franke et al., 2007; Goos, 2004). Norms that 

support students in feeling comfortable sharing their ideas and taking risks might include regular 

opportunities for open discussion, upholding the notion that learning is more important than 

performing, and valuing perseverance and revision of thinking more than immediate correctness 

(Boaler, 2015; Jansen, 2020). Part of a conducive classroom environment can include a 

pedagogy of care, which is characterized by pedagogy that shows students concern for both their 

  23 

academics and well-being, and this is built through strong relationships and trust with and 

between students (Goralnik et al., 2012).  

The outer circle in the framework presented in Figure 3 is Equity Goals, which 

everything the teacher does with the task, how it gets enacted, and how the classroom culture is 

established, becomes situated within or is driven by. Because “equity” is a complex and 

multifaceted word, I draw on Gutiérrez’s (2009) framework for dimensions of equity (1) to 

communicate what I mean by equity and (2) to frame ways in which justification opportunities 

might advance equity. Gutiérrez describes four dimensions of equity: access, achievement, 

identity, and power. Access involves having resources and opportunities that allow students to 

fully participate in mathematics, such as high-quality curricular tasks designed to elicit 

justification. Achievement refers to institutionally acknowledged performance and participation, 

such as students’ participation in rigorous mathematics classes that involve high order thinking 

such as justification. Identity means students seeing themselves reflected in and as doers of 

mathematics. Power is characterized by student empowerment, voice, and influence in the 

mathematics classroom and beyond. Gutiérrez describes access and achievement as part of a 

“dominant axis” and identity and power as comprising the “critical axis.” Depending on a 

teacher’s goals for using justification, this practice may be implemented in different ways and 

potentially with a focus on different dimensions of equity.  

Methodology 

Research Questions. The research questions guiding this study focused on understanding the 

opportunities for mathematical justification that one teacher provided students in her classroom 

and the impact she perceived these had on equity for her students. The first research question 

was, How does a teacher facilitate opportunities for students to engage in the mathematical 

practice of justification? The second research question was, How does the teacher perceive 

mathematical justification as supporting her equity goals? 

Study Design. This research applied case study methodology, which can be described as a 

detailed description and analysis of a bounded system, where the parameters of the study are 

bounded by the “case” (Merriam & Tisdell, 2015). In this study, the teacher’s practice, 

dispositions, and perspectives were the “case,” and these were explored through interviews, 

classroom observations, and analysis of implemented curriculum materials. This study was 

situated in Merriam’s (1998) constructivist pragmatic epistemological assumptions of case study, 

  24 

meaning findings might be applied to offer insight into ways other teachers may consider 

providing similar opportunities, as adapted within their own contexts. The goal of this study 

methodology was to understand and amplify the practices and perspectives of one teacher and 

how she provides opportunities for mathematical justification in ways that centered students’ 

ideas and voices. 

Sampling and Participant Background. The participant in this study was one teacher, Josie 

(her chosen pseudonym). The selection of this teacher drew from a combination of convenience 

sampling methodology due to proximity and my pre-established relationship with the teacher and 

purposeful sampling methodology based on the teacher’s existing practices that align with the 

study (Merriam & Tisdell, 2015). For instance, when considering participants for the study and 

speaking with Josie about it, she mentioned that she starts every lesson with a Number Talk 

(Parrish, 2014) and that she often incorporates justification opportunities in her lessons. 

Additionally, it was ideal that Josie and I already had an existing relationship, having both taught 

at the same school at one point. Because the teacher and I already established a relationship from 

formerly working together, this provided a strong foundation for a trusting research-participant 

relationship.  

At the time of the study, Josie had nine years of teaching experience, primarily in self-

contained, upper elementary classrooms, teaching students of racially/ethnically diverse 

backgrounds and in low socioeconomic regions. She also had some experience teaching middle 

school mathematics. During the study, Josie taught fourth grade. Josie identifies as a Filipina 

woman and immigrated to the United States from the Philippines at age 13. From once working 

in the same school with Josie, I knew she had a strong commitment to addressing issues of social 

justice, reflecting students’ identities in the content, and incorporating student voice and agency 

into the classroom. This was Josie’s sixth year at the school.   

School Context. Josie’s classroom was situated within an urban, public charter school. The 

school, Midwest Academy (pseudonym chosen by Josie), was known for its commitment to 

social justice and place-based education. The school hired teachers who shared this mission and 

required teachers to incorporate place-based projects into their curriculum. In describing what 

place-based learning means, in the context of her school, Josie explained,  

[Midwest Academy] defines it as creating critical thinkers who contribute to the wellbeing of 

their community. I guess that place based is like putting the learning in the context of your place. 

  25 

 
Sometimes you can solve problems in that place that you know can come from your learning, or 

sometimes it's just you can learn about your place to develop and foster love of it and 

stewardship of it. 

The majority of the students at Midwest Academy were from groups that are currently 

and historically marginalized, including Black and Latinx students. The school is designated a 

Title I status, and because the majority of students qualified for free or reduced lunch, the school 

provided all students free breakfast, lunch, and snacks during the school day. 

Researcher Positionality. Given that I formerly taught at this school and had a pre-established 

relationship with Josie, I brought a somewhat “insider” lens to this context. For instance, I knew 

that this school and Josie had a strong commitment to place-based education and social justice, 

and that integrating these within their existing mathematics curriculum was sometimes more 

challenging than other subject areas. I also knew that new teachers at the school were typically 

sent to a professional development on Responsive Classroom (Responsive Classroom, 2020) 

prior to the start of the school year, and that a major focus of the beginning of the school year in 

Josie’s classroom was on responsive classroom routines. While I was at the school, I did not 

collaborate in lesson planning with Josie nor observe her instruction, which means the lens I 

brought toward her specific classroom verged toward the “outsider” lens. This insider/outsider 

perspective I brought suggested to me that Josie’s classroom would provide an interesting 

context to explore how justification opportunities could be provided, and particularly, in ways 

that might advance critical dimensions of equity. I recognize I brought to this research a 

particular status, as a White researcher from a prominent research university entering the space 

of a classroom of predominantly Students of Color and a Teacher of Color. Because I knew my 

presence could have some influence on the space, it was important to me to work with a teacher 

with whom I had an established and trusting relationship, who knew me as a teacher at one point 

as well, and where my role was to learn from her. Mindful of my status, I was conscientious 

about decentering myself by quietly observing while in the classroom and centering Josie’s voice 

during our meetings, by asking questions to position her as expert and by not offering or 

answering questions about my opinions.  

Data Collection and Analysis. Data sources included recordings and transcripts of interviews 

and lesson planning meetings, field notes of classroom observations, and implemented 

curriculum materials. The first classroom observation was conducted in November 2023 to 

  26 

gather a general idea of Josie’s practice and routines established that school year, and to inform 

the development of pre-interview questions. A two-part pre-interview (totaling 130 minutes) was 

conducted in January 2024 to gather background information (e.g., about Josie’s identity and 

teaching experience) and Josie’s initial thoughts on justification and equity. For example, I asked 

Josie, “What relationship, if any, do you see between equity and the practice of justifying?” 

Regular classroom observations occurred two days per week (typically consecutive) from mid-

February through April 2024, totaling 16 observations, in which I took field notes. Field notes 

were taken on types of student justifications (e.g., justifications of their ideas, claims, choices, or 

mathematical relationships) and on Josie’s justification implementation (including pedagogical 

strategies and norms for eliciting or promoting justification, such as discourse moves or 

statements that decentered herself as the mathematical authority). Classroom observations began 

approximately halfway into the school year in order to examine how justification opportunities 

would be facilitated well-after routines and norms were established by Josie in the classroom, 

and prior to any influence that me entering the space as an observer could have.  

Dispersed throughout the classroom observations were six (30-60 minute) lesson 

planning meetings, where Josie debriefed past lessons and shared plans for upcoming lessons. In 

these meetings, Josie debriefed the past week’s justification opportunities, reflected on questions 

I asked about observed lessons in relation to justification and her equity goals, and shared her 

curriculum and instruction plans for the upcoming week. To facilitate lesson debrief 

conversations, I asked Josie to reflect on the justification opportunities that played out during the 

previous week, including what she thought went well and what challenges she experienced. I 

asked Josie what she noticed about students in relation to her equity goals across those lessons 

and how she was considering those equity goals as she planned for the upcoming week. 

Sometimes I asked questions to understand more about something I observed or about 

occurrences earlier in the school year. For instance, in order to better understand the types of and 

reasons for the Number Talks Josie created, I asked, “You also did a Number Talk related to 

something that you were doing in class, which was your Read Harder Challenge. Have you done 

other Number Talks outside of these two– the Loyalist-Patriot Debate Number Talk and Read 

Harder Challenge Number Talk–  that have also related to other things happening in your 

classroom?” This was followed up with the question, “Why are these types of number talks 

important to you?”  

  27 

Josie uploaded all implemented curriculum materials during the February through April 

observation window (which included the materials from lessons I observed and those I did not 

observe) into a shared folder. Viewing the curricular tasks implemented before and after lessons 

I observed helped to provide a better understanding of curricular coherence and how Josie’s 

lessons built toward supporting her equity goals during different stages of a curricular unit. 

Keyword analysis and thematic analysis were conducted on the design of all curricular tasks she 

implemented during the observation window. Keyword analysis was used across these 

curriculum materials to identify words and phrases related to justification, such as identifying 

problems in worksheets that included words such as why, prove, true, explain, notice, and 

pattern. Thematic analysis (Braun & Clarke, 2006) was used to group tasks by themes (such as 

tasks that were designed by Josie for cultural relevance, tasks involving patterns, etc.) to look for 

potential connections for supporting the equity goals she expressed. For example, cultural 

relevance was one theme that emerged. A task that showed cultural relevance included mention 

to students’ experiences, interests, or community (e.g., a task that referenced the amount of 

rainfall in the students’ city) because of the connection to their community. Mathematical 

patterns was another theme that emerged during thematic analysis of the curricular tasks. One 

example of a task grouped into this theme included the direction, “Describe a pattern that you 

notice is true about multiplication for numbers that end in zeros.” Within this task about noticing 

patterns was a connection to agency because students were being tasked with developing a 

conjecture about what happens when numbers are multiplied by powers of ten, whereby students 

are assuming ownership of the mathematics since the teacher is not imparting the mathematical 

rules or procedures.   

In May, an exit interview (60 minutes) was conducted to gather Josie’s final reflections 

and more deeply examine questions around justification and equity that emerged during the 

study. For instance, since Josie reported the beginning of units to be important places to leverage 

the practice of justification for supporting her equity goals, I asked, “In what ways do the 

beginning of units or new topics lend themselves well to incorporating opportunities to justify?” 

To help me summarize some of her strategies and perspectives, I also asked Josie what advice 

she would give to other teachers looking to provide more justification opportunities. I probed to 

ask about how another teacher should go about designing tasks as well as what strategies or 

  28 

questions the teacher should ask to support students’ own inquiry, claims, and reasonings along 

the way.  

Transcripts (of interviews and lesson planning sessions) and field notes (of classroom 

observations) were also analyzed using a thematic analysis approach to understand Josie’s 

practice and perspectives around justification and equity concepts. This analysis involved 

reading all field notes and transcripts to identify quotes that could help explain or elaborate each 

circle of Figure 3, and organizing these quotes into themes. The same process used to identify 

themes from quotes as described earlier for curricular tasks (which is circle 1) was applied for 

each circle of the framework in Figure 3. This involved creating a separate data table for each 

circle of Figure 3. For the first circle (the innermost circle), Written Task, keyword analysis and 

thematic analysis (as described earlier) were conducted to look for patterns in the types of tasks 

Josie designed and selected to implement. For the other circles, thematic analysis was conducted 

on field notes and transcripts (of interviews and lesson planning meetings) separately for each 

circle, resulting in multiple rounds of analysis of the same data, but with attention to each circle 

separately. For the second circle, thematic analysis was conducted to look for themes in Josie’s 

justification-oriented task enactment. Themes included activating background knowledge, 

pressing for reasoning, involving peers, and sharing in authority of knowledge. For example, in 

one lesson, Josie pressed for one students’ reasoning to elaborate on why they thought the 

answer was wrong. Her statement, “What makes this wrong? Why does it all need to be single 

digits?” is eliciting a justification for an answer a student is disagreeing with. For the third circle, 

analysis involved looking for themes around how Josie provided a conducive classroom 

environment. References to math norms and statements around supporting students in feeling 

comfortable or confident to participate were themes coded for under the conducive classroom 

environment category. For the fourth circle (the outermost circle), thematic analysis was used to 

look for themes around explicit connections to the equity goals expressed by Josie. The goal of 

examining themes within each circle of the framework in Figure 3 was to examine the multi-

system nature of justification opportunities facilitated by Josie and her intended impact on equity 

for students.  

Results 

In this section, findings are shared through the lens of the framework in Figure 3. First, 

examples from Josie’s classroom and perspectives that elaborate each circle of the framework 

  29 

are shared, and then an example that cuts across the circles is discussed. Any quotes where the 

word, like, was interjected as a filler and not needed for context, was sometimes removed for 

readability. All names used are pseudonyms.  

Task Design. Josie designed most of her curriculum materials herself, reporting that her primary 

curriculum “doesn’t suit the kids at our school.” She described the primary curriculum material 

as heavily procedural, overly scaffolded, not inquiry-oriented, and not culturally relevant, which 

is especially important to her working in a school focused on place-based learning. According to 

Josie, creating materials herself allowed her to connect content to the students and to their 

community, and opened spaces for justification opportunities. Keyword analysis revealed that 

tasks included references to their city (e.g., examining the amount of local rainfall), names of 

students in the class, interests held by students in the class (e.g., calculating the number of bottles 

of glue needed to make slime for a given amount of friends), and references to other events 

happening in their classroom or school (e.g., calculating the percentage of their class who have 

completed their reading challenge thus far).  

Key terms that surfaced in tasks for eliciting justifications included words such as 

pattern, noticing, true, prove, why, explanation. Phrases included: Using what you know about…, 

Find the patterns or make up a rule for…, Describe a pattern that you notice is true about…, Use 

at least X strategies. None of the tasks included the word “justification” or “justify.” When asked 

why she used the word “explanation” instead, Josie said that she thought it was a more accessible 

word to her students. However, Josie intentionally incorporated correlated key words in tasks to 

signal to students that she was looking for a justification. For example, “Write (in words) your 

explanation for why this is a true equation. Is your explanation true for other examples? Give 

two equations that are similar to the example that follow your explanation.” Here, the word 

“why” signals justification, and the request for multiple examples associated with the proving of 

something to be “true” could denote a way to support or enhance the justification.  

Justification-oriented Task Enactment. Some of the tasks Josie created did not include 

directions for justification, but justifications were elicited through her enactment strategies. 

These included pressing for reasoning, involving peers to build from one another’s ideas, 

activating background knowledge to elicit a conjecture to explore, and sharing authority of 

knowledge with students.   

  30 

In describing how to elicit a justification, Josie explained that it required a lot of 

prompting to get at students’ reasoning. She elaborated, “But definitely the why requires more 

prompting. ‘Cause it is, you know, like deeper thinking. Like you really have to notice whatever 

it is, the pattern that you're trying to explain, and that sometimes takes time for most people.” 

When asked what types of prompts she used, Josie said, “Maybe I'll ask them something specific 

like, Why is there always this number here? or Why did you know you could do that specific 

‘whatever’ operation? Or it's just, I guess it depends on the ask. But just very specific questions 

about what they're working on.” Some questions that I observed Josie ask to elicit justification 

during math lessons included, “Why did you care about making it equal?”; Do you want to talk 

about why you added the values?”; and “How does that make sense?” 

Another enactment strategy used by Josie to facilitate justification was involving peers. 

In describing why she did this (and referencing a student from class whom I frequently observed 

ask questions), Josie explained, “Maybe some kids who are too afraid to ask are then learning 

from that other kid who's just gonna ask why. Anyways, she's great to have in class, because 

she's like, I don't understand. Why is it?" Sometimes Josie relied on other students’ words to 

justify concepts to their peers in a way she was not able to or asked them to rephrase in their own 

words what they understood their peers did. She shared,  

Honestly, she helped me. She was like, she said it in a way that I never would have said it. And I 

was just trying to repeat it in my own words… Sometimes I ask other kids to explain what they 

understood about what somebody said. 

Josie explained that sometimes another student’s justification is what can click for 

someone and “if there's twenty-four different people sharing a perspective, there's more chances 

that that might happen.” Persuading a peer is another strategy she used to elicit justification. For 

example, in a lesson I observed involving equivalent fractions, Josie asked, “Did you persuade 

the other person that it is or isn’t equivalent?” 

Part of Josie’s pedagogical approach was supporting students in noticing patterns or 

relationships and coming up with “rules” themselves based on their inquiry, which involved 

activating students’ background knowledge to get them to generate a conjecture to be explored 

and justified. At the beginning of a lesson I observed on equivalent fractions, Josie began by 

telling the class what they would do that day and reminding them where they left off the previous 

day, “We’re going to come up with a rule for equivalent fractions today. [Yesterday], you used 

  31 

the fraction bars to create equivalent fractions. The last question asked you to notice a pattern 

and write a rule, but we ran out of time.” Josie summarized what three students did yesterday, 

“Aniah found 1/8 is the same as 2/16,” and then Josie drew the corresponding fraction bars 

beside it. Josie continued, “Jared noticed 1 times 2 is 2, and 8 times 2 is 16. Jess saw 9/12 is the 

same as ¾. She said you multiply the top and bottom by the same number. Is this always true? 

Does it work for every fraction? Talk about it with your partner.” After taking time to talk this 

through with a partner, Josie gathered the class back together. The discussion led to the 

following dialogue, which discussed multiplying by odd and even numbers:  

Josie: Could I get from 1/8 to 4/32 without knowing 2/16? How? 

Student: Knowing we’re doing it by 2s, we can also do it by 4s, 6s, and 8s. 1 times 4 is 4, and 8 

times 4 is 32. 

Josie: Does it also work for times 10, 12, 14? Does this only work for even numbers? You said 

3/6 is one-half, one time. You said 5/10 is one-half. You were very certain of that. What about odd 

numbers? 

Student: You can still multiply by odd numbers to get another equivalent fraction. 

Josie: What do I multiply ½ by to get 5/10? 

Student: Multiply the numerator and denominator by 5. 

Josie: Is that true? 

Student: Yes, because 1x5 is 5 and 2x5 is 10. 

Now, Josie felt ready to ask the class, “Who can make up a rule for equivalent fractions 

knowing what we know?” One student answered, “If you multiply the numerator and the 

denominator by the same number, it will stay equivalent.” Josie recorded this on the board and 

then said to the class, “I’m going to check with these problems on our worksheet and check if 

that’s true” and proceeded to talk through that process with the class, to conclude the exploration 

of this math rule for equivalent fractions that the class generated and tested together. Given that 

time was running short at the end of class, it makes sense that Josie facilitated this last step in a 

more teacher-centered way quickly, but in a way that wrapped up the ideas that originated from 

the students. 

An overarching theme of Josie’s pedagogy and justification enactment strategies was 

sharing mathematical authority with students. While this surfaced in the enactment strategies 

previously described, I elaborate some more of her thoughts and examples here. It was important 

to Josie that she was “not the knower of everything,” as she told me. She tried not to give 

  32 

answers to students, stating, “I very rarely try and give the answers to them, ‘cause, you know, I 

think I've said this before, but I don't want to be like the keeper of math. I want them to be the 

thinkers.” Through her pedagogy, Josie endeavored to send the message to students that they 

were doers of mathematics and should feel empowered to have the authority and agency to make 

discoveries and claims, and to justify those themselves. Similarly, Josie purposefully did not 

answer questions sometimes, explaining, “I do it pretty frequently, so they know if I'm not 

answering a question, they're gonna stop asking because I want them to work through it.” 

Incorporating daily discussion and inquiry into instruction served to shift authority of knowledge 

to the students. Josie shared, 

I love, you know, doing math discussions, and I think that engages the kids, too, to feel like they 

can share ideas, you know, and not just like, “Hey Miss. [Josie] told me to do this specific thing 

for this problem.” But they could be like, “I noticed this pattern. What if we tried this pattern 

across these other things, will we see the same thing? And then they ask, “Well, why didn't we- ? 

Or why did we- ?” So we always start with discussion, and that's built into like every day for us, 

too.  

Building discussion into everyday practice normalized for students that mathematics is 

about exploring, sharing, and justifying one’s own ideas and learning from peers doing the same. 

Conducive Classroom Environment. Justification opportunities were importantly situated 

within a conducive classroom environment, where students felt safe sharing and taking risks 

(Klosterman, 2016). For Josie, establishing a conducive classroom environment involved co-

creating classroom agreements, building students’ confidence in mathematics, promoting their 

curiosity, and humanizing the mathematics learning experience so that students could feel 

comfortable sharing ideas (including uncertain or in-progress ideas) and taking risks (such as 

disagreeing with peers who may often get correct answers early on). 

 When Josie was asked why she thought her students were so comfortable sharing their 

ideas even when they might be wrong or disagreeing with their peers, she referenced co-creating 

math agreements together at the beginning of the school year and establishing these as norms 

throughout the year, 

I think setting the norms and making the agreements together first… So we were like, everyone is 

going to feel like they contribute to this classroom and you're gonna be a person that helps in 

that. You know what I mean? Not prevents people from their own learning. And I think they 

  33 

understood that. So, I think part of it was the kids buying in and willing to be those kind of people 

for each other. 

Part of the math agreements were that each student had a responsibility to contribute their 

ideas and also a responsibility to support others in their efforts to contribute, which meant 

uplifting the knowledge that students brought to class and being intentional about building their 

confidence and sharing their mathematical thinking with others. Josie explained, “Everyone 

came to the classroom with a tool belt of knowledge. And they can share whatever is in there, 

and we can add to it and everyone can contribute. So I think that is part of agency.” Emphasizing 

this from the start meant boosting the confidence of more reserved students early on. Josie 

mentioned, “For some kids, especially kids that aren't as comfortable, you have to really find 

those moments… like taking their notebook and showing the class, like look at what this person 

did. And I think that takes a lot of deliberate teacher action at the beginning.” These agreements 

developed into norms, as they were not just written at the start of the school year but, as I 

observed, carried deep into the school year. Josie elaborated, “It has to be a culture. It has to be 

within this space, a space where I feel safe to share an idea and be okay with being wrong or 

being celebrated for being.” Sometimes Josie modeled for students when she made mistakes and 

how to respectfully disagree with a peer. She shared, “I practice also, a lot like, telling the kids to 

disagree with me... I think it helps them understand that they're all capable, not just that an adult 

can tell them something, but that they can see sometimes that, you know, adults slip up, or 

anybody, and that there is a kind way to engage with somebody and help them understand 

something, and don't agree with them.” This also contributed to Josie’s efforts to position 

students as capable and share in authority of knowledge. Josie iterated, 

I think a student that sees themselves as capable they’re more willing to share their thinking… I 

think that that is more important than me being an authority of knowledge in a classroom, and as 

part of learning, because I cannot give my knowledge to somebody. I can’t. That's never– you 

know what I mean? If I could, my job would be so easy. 

Supporting students’ ease and comfort in the mathematics classroom involved inspiring 

students’ curiosity to want to explore and share their noticings as well as to develop an overall 

joy for and comfort in doing mathematics. Josie shared, “I just think if a child or person is more 

curious than afraid, they're more willing to try. So I think that helps them access it, if they're 

willing to try.” Multiple times, she shared that one of her top goals as a mathematics teacher is 

“promoting a curious mind” and “engaging a curious mind.” For Josie, a lot of joy can come 

  34 

from being curious. She explained, “There's so much joy coming from being curious, you know, 

and to want the desire to learn, the desire to find out about something you just decided to be 

interested in because of whatever reason. I don't know what would happen to somebody that lost 

that.” For Josie, promoting curiosity was more important than speed and immediate correctness, 

which contributed to her conducive classroom environment. She believed, “If you understood 

that you could be curious and ask questions and explore something instead of having to know it 

right away. But then you, there's a lot of self-confidence that has to be built with a kid that sees 

math and problems that way.” For Josie, she promoted curiosity and reinforced for students it is 

okay to not always know the answer, (disrupting the math anxiety associated with traditional 

mathematics learning), in order to build students’ confidence and comfort to participate in the 

justification process.    

Josie facilitated a justification-oriented conducive classroom environment so well that 

students felt comfortable disagreeing with their peers who were perceived to be the “math whiz” 

of the class, and even these “math whiz” students felt comfortable sharing when they were not 

certain of their correctness. After observing a lesson where this was very apparent, I asked Josie 

about it and she explained, 

I know that the kids perceive Jonah especially, even since second, third grade, because he's been 

in Faith’s (pseudonym for Josie’s daughter) classes like the math whiz or whatever. He's just 

really natural with it. He understands patterns really well. And so I know the kids kind of see that. 

But even sometimes, and I'm like, don't want them to always go with what they say, you know. 

And sometimes they don't, which I'm like, okay so they're still thinking for themselves, and they 

perceive not Jonah as the keeper of math, but all of them as able to do math, and they can 

disagree with even somebody that, like Jonah, who is often right. 

Describing another example with a different “math whiz” in the class, Josie shared, 

“‘Cause sometimes in math what's intuitive isn't always, you know, the answer or something. 

And sometimes you didn't– Evan will get something that is, like seems intuitive, like it should be 

this, and it's not. And so there's been instances where that's happened.” Contributing to Josie’s 

conducive classroom environment was students seeing instances in which the “math whizzes” 

were also offering answers with justifications that ended up being incorrect.  

Both Jonah and Evan were White boys in a class of predominantly Students of Color, so I 

asked Josie how she thought other students thought about Jonah and Evan both being White boys 

and seemingly intuitive with math. Josie stated that the “culture around race and gender is also 

  35 

 
really different than other schools that I've experienced.” I asked her to say more about that and 

she explained that “identity is so celebrated at our school,” sharing how they make identity 

wheels at the beginning of the year and that kids are open about their pronouns, including kids 

who identify as “they.” Josie also shared about a female Student of Color who presented to 

others as being great at mathematics and how tracking who shared during Number Talks 

provided opportunities for students to see their peers of all genders and races (particularly 

marginalized identities) showcase their mathematical smartness. 

I guess we talk about race pretty openly and gender pretty openly… I've had students that are 

nonbinary, that are openly that, and openly are like, I prefer ‘they’... I think identity is so 

celebrated at our school. We make identity wheels at the beginning of the year. We talk about 

inclusivity all the time, that even though it happens to be that the two, two of the kids in the class 

that are really vocal about their method are White boys. There are also a lot of examples of like 

Amina. She's Mexican or no, she's Colombian and Arabic. Half and half. And she's really great at 

math. So there's like other examples of identities that they see that are super capable at math as 

well, but maybe do it more quietly, or do it less boisterously or something. But they still have 

good shares. And because I think Numbers Talks is set up the way that it is, we're like, you can't 

share every day, like a different person, different three people are highlighted every day. They 

still see those identities as people doing that. So I think it's just culturally at our school, we just 

always talk about identities and how there's value in every identity. And maybe a lot of them have 

been going to that, just that school, since kindergarten. Maybe they've never lived in a world 

where the perception is like, you know, men are good at math and women aren't, or something. 

Josie believed that the reason students in her classroom (which were predominantly 

Students of Color) seemed comfortable disagreeing with math whizzes, Jonah and Evan (who 

were White boys), was attributed to (1) there being a culture of inclusivity and discussion of 

identity school-wide and (2) seeing regular examples of girls, non-binary students, and Students 

of Color showcasing mathematical smartness during lessons in her classroom. 

Disposition for Leveraging Justification Opportunities to Support Equity Goals. Josie 

perceived it possible for justification opportunities to advance all four dimensions of equity 

(access, achievement, identity, and power) (Gutiérrez, 2009). Gutiérrez (2009) states that 

addressing all four dimensions is required for fully realizing equity. Because Josie believed 

justification opportunities can advance each dimension, and because she saw each dimension as 

intertwined with the others, this suggests a correlation for Josie between justification 

  36 

opportunities and equity advancement. Shared below are Josie’s thoughts on the relationship 

between justification opportunities and each of these dimensions of equity. Furthermore, for 

Josie, a focus on facilitating students’ agency and cultivating students’ development of positive 

mathematical identities emerged as prominent themes in her practice and perspectives, 

suggesting these to be critical constructs for her in influencing the other equity dimensions. I am 

using the equity construct of agency to mean students being empowered to have influence in 

their learning environment, by which they play a role in their own mathematical development 

and when they learn from/with their peers through sharing authority of knowledge with the 

teacher (Grootenboer & Jorgensen, 2009). By positive mathematical identity, I refer to students 

seeing themselves as doers of mathematics and perceiving confidence in their capabilities to 

engage in rigorous mathematical practices (Darragh, 2013).  

Because “equity” is a word that is multifaceted and conceptualized in nuanced ways 

across the literature and school contexts, I provided Josie a brief description of each dimension 

of equity as defined by Gutiérrez (2009), to help guide our conversation in a way that provided 

for shared language around the word. After sharing Gutiérrez’s conceptualization of equity, I 

asked Josie, “Where do you see the practice of justifying within each of these dimensions?” She 

believed, “Justifying supports all of those things.” Josie discussed each dimension as intertwined 

with another dimension, interestingly aligned with how Gutiérrez frames them. Regarding the 

relationship between justification and identity, Josie mentioned when you’re justifying, “the 

reasoning is your own” and “you're able to voice your own critical thinking of whatever it is.” 

With power, her perception was, “You did the math yourself, so that part is like identity. And it's 

part of power because that's, again, centered around yourself… You do the task without, I guess, 

being trapped in an algorithm. You don't have to follow the step-by-step process right now.” 

Josie explained that attending to dimensions of identity and power (which have connections to 

agency), in turn, promoted access and achievement. Referencing the achievement dimension, 

Josie stated, 

I think all of that probably leads to the achievement part where kids are more open to 

participating, because nobody, they don't feel maybe the pressure of the correct strategy or the 

correct thing that it should look like. They can talk it through. And that they're told it's okay to 

have an unfinished thought, even to share that out because there’s still value in that. 

  37 

Similarly, Josie shared that focusing on mathematics as a justification process can make 

mathematics more accessible to students because it removes the pressure of having to produce a 

correct final product (e.g., focus on correct answer) and students can be creative in their 

problem-solving (reflecting their mathematical identities and agency to do so),  

I think that thinking of math as a process makes it more accessible to a lot of kids. Makes it 

friendlier, makes it feel maybe less pressure. Math isn't just you show up, you do the thing, and 

you get the answer. You're like allowed to be more creative with it. And I think a lot of kids do 

well with creative tasks, because they can kind of put themselves into it.  

When justification was a norm situated within a conducive class environment where 

students did not feel pressured to do mathematics in a constrained way, students’ agency and 

identity were cultivated. Talking about both, Josie explained,   

Sometimes they come, and they are so hung up on the correct answer, and I think for them to 

discover that they can explain their thinking, and that they can logic through stuff... I think just as 

part of their identity, perceive math to be like, ‘Oh, it's something I can think about and actually 

do. And there's, I'm not stuck in one way, and if I don't know the way I'm just instantly wrong or 

dumb, or whatever.’ I always tell them, ‘Look at it first. Use what you know to figure out what 

you don't. You can see things. You came into this classroom already knowing things.’ 

Josie saw that all students brought a wealth of knowledge, experiences, and perspectives 

with them that they could apply to their solving of problems, and Josie saw justification as 

opening opportunities to reflect students’ identities in the learning process. It was important to 

Josie that students did not get fixated on mathematics just being about correct answers, but that 

doing mathematics meant discovering, problem-solving, and explaining your thinking using what 

you know. When I asked Josie directly about the relationship between agency, identity, and 

justification, she rhetorically asked,  

Don't they kind of go hand-in-hand? Or to me, it feels that way. Like, isn't the opportunities to 

have their thinking be heard and valued and uplifted, is kind of the agency part? I think that 

justification opportunities and conversations inherently provide kids with agency.  

Josie saw cultivating students’ agency and identity as intertwined. These were the 

primary equity goals Josie had for her students inside and outside of the math classroom, and she 

leveraged mathematical justification to support these goals. By doing so, Josie believed she was 

influencing other dimensions of equity, including access and achievement. Being a facilitator of 

students’ agency in the learning process and their positive mathematical identity development 

  38 

was important to Josie’s goal of cultivating free-thinking students inside and outside of the 

mathematics classroom. Josie believed, “Because really, that's where we're leading them to is 

like to be outside of our walls and be free thinking humans out in the world.” 

For Josie, sharing authority of knowledge with students was particularly supportive for 

facilitating their agency and development of positive mathematical identity. This was evident 

through her practice and perspectives. As previously shared, Josie expressed, “I very rarely try 

and give the answers to them, ‘cause, you know, I think I've said this before, but I don't want to 

be like the keeper of math. I want them to be the thinkers.” This is why it was important to Josie 

to provide students opportunities to discover and justify mathematical patterns/relationships 

themselves in ways that centered their ideas and working understanding of content. According to 

Josie, supporting students’ agency and positive mathematical identity development ultimately 

can support access to content and achievement in mathematics, and justification is a practice that 

can be leveraged to promote these goals.  

Summarizing Back the Framework. For Josie, leveraging justification opportunities to support 

her equity goals involved purposefully designed curricular tasks, such as those that involved 

students in discovering and justifying mathematical patterns or relationships themselves whereby 

they took ownership of the mathematics. Engaging students in justification (in general, and 

particularly in ways that met her equity goals) involved more than just a purposefully designed 

curricular task. As illustrated in Figure 3, what drove how and why a given justification task was 

enacted were Josie’s dispositions for critical equity constructs. For Josie, her primary equity 

goals of supporting students’ agency and positive mathematical identity development meant 

implementing justification in ways that shared authority of knowledge with students. This more 

active student role required a conducive classroom environment where students felt comfortable 

sharing their ideas and taking risks.  

Example that Moves through the Framework. Here, I use the framework shared earlier (i.e., 

Figure 3) to help illustrate how Josie’s tasks were embedded within particular enactment 

strategies, which were embedded within a conducive classroom environment, and driven by her 

equity disposition to support students’ agency and development of positive mathematical 

identities, in order to facilitate equity in her classroom. In this section, I elaborate one of Josie’s 

tasks to illustrate how that task and the multiple systems represented by the framework were 

  39 

embedded within one another in order to support equity. As described earlier, one task Josie gave 

students was to develop a rule for finding equivalent fractions. 

Figure 4. Written Task 

For Josie, enacting this task in ways that met her equity goals meant sharing authority of 

mathematical knowledge with students. This involved first activating students’ background 

knowledge, which Josie did when she reminded the class about the equivalent fractions that 

Aniah, Jared, and Jess had noticed. Josie also asked prompting questions and pressed for 

reasoning. She asked students, “Could I get from 1/8 to 4/32 without knowing 2/16?”; “Does this 

also work for even numbers?…What about odd numbers?”; and “Is that true?” These types of 

questions supported students in thinking for themselves and needing to justify the mathematics 

behind their claims and ideas.  

  40 

 
 
Figure 5. Written Task, Situated in Task Enactment 

This task and its enactment were embedded within a conducive classroom environment. 

Josie provided students time to explore over the course of multiple days. She positioned students 

with authority of knowledge, to build their confidence so that they could feel comfortable 

sharing their in-progress ideas. This lesson took place in April, so the math agreements had 

solidified into norms, where students learned that part of the learning process was exploring and 

justifying claims, which meant not having answers right away and raising potentially incorrect 

ideas was okay. 

  41 

 
 
 
Figure 6. Written Task, Situated in Task Enactment, Situated in Conducive Class Environment 

The written task, as embedded within Josie’s enactment, as situated within the conducive 

class environment she fostered, were all driven by Josie’s equity disposition to support students’ 

agency and positive mathematical identity development. This is why it was important to Josie to 

let students come up with a mathematical rule themselves rather than her imparting the 

mathematical relationship. This involved giving students the agency to explore, discover, and 

justify mathematical ideas in ways that made sense to them and reflected themselves in the 

learning process. For Josie, this built toward her goal of wanting students to be free thinking 

humans inside and outside of the mathematics classroom. 

  42 

  
 
Figure 7. Written Task, Situated in Task Enactment, Situated in Conducive Class Environment, 
Situated in Equity Disposition for Supporting Agency and Positive Mathematical Identity 

Discussion and Conclusion 

Given how infrequent justification is incorporated into classrooms and how challenging it 

is for teachers to notice and implement the practice (Melhuish et al., 2020), it can be said that 

incorporating justification would be complex work and that teachers would need to view the 

practice to be relevant to their context and equity goals in order for them to implement it. To 

better understand how and why one teacher who was committed to regularly incorporating 

justification opportunities perceived justification as a practice that can be leveraged to support 

her goals for advancing equity, this study explored the questions, How does a teacher facilitate 

opportunities for students to engage in the mathematical practice of justification? and How does 

the teacher perceive mathematical justification as supporting her equity goals? Examining 

Josie’s facilitation of justification opportunities illustrated the complexity of this work because it 

involved attention to multiple factors situated within one another (i.e., purposeful task design, 

situated within justification-oriented enactment, situated within a conducive classroom 

environment). Furthermore, Josie’s attention to these factors were driven by her goals for 

facilitating students’ agency and development of positive mathematical identity in order to 

support equity. According to Josie, justification can advance all four dimensions of equity 

  43 

 
 
(access, achievement, identity, and power), which Gutiérrez (2009) describes all as intertwined 

and required for realizing equity. It makes sense that focusing on facilitating students’ agency 

and positive mathematical identity development was possible through justification, given that the 

practice opens spaces for centering students’ voices and perspectives through their mathematical 

claims and reasonings rather than using and practicing the teacher’s mathematics. It also makes 

sense that supporting students’ choice and voice in the learning process (agency), and supporting 

students in seeing their ideas and perspectives reflected in the learning process (identity) can 

support access to mathematical content and achievement in the subject. When students feel 

connected to the content and are actively engaged with the content, these can facilitate 

opportunities for access and increased participation patterns and achievement. However, more 

research is needed to understand the relationship between students’ agency in the learning 

process, positive mathematical identity, and mathematical achievement. Fernandez et al. (2024) 

is one study that shows a beginning understanding of this relationship, finding a statistically 

significant relationship between a student’s positive mathematical identity and achievement, with 

pedagogy that centers students’ reasonings as a contributing factor. In describing this 

relationship, Fernandez and colleagues (2024) explained, “students who see themselves as 

individuals who can do and practice mathematics tend to perform better on mathematics tasks, 

even net of multiple other covariates of both identity and achievement” (p. 2018). Though more 

research is still needed to examine the role of justification in advancing equity (including 

developing a better understanding of the relationship between mathematical identity and 

achievement), it does make sense that pedagogy which leverages mathematical reasoning 

practices where students can see themselves as the “authors of mathematics” (Fernandez et al., 

2024, p. 2017) can play a key role in supporting students’ mathematical identity and 

mathematical achievement. 

 Leveraging justification opportunities could be particularly conducive to supporting 

equity in classrooms of racially/ethnically diverse students, whose voices and perspectives have 

and continue to be marginalized. Not all students perceive themselves as doers of mathematics, 

or see or know people who look or identify similar to themselves as doers of mathematics. 

Advancing equity means being intentional about supporting students of marginalized 

backgrounds to see themselves as capable mathematics doers, who have choice and voice in the 

learning process, and see themselves reflected in the learning process. This study suggests that 

  44 

justification could be a powerful practice that can be leveraged to facilitate equity, though more 

research is needed to understand the impact on students’ perceptions and experiences since the 

unit of analysis in this study was the teacher. Examining the student experience could involve 

administering surveys and conducting focus groups or interviews with students, to look for 

potential alignment between the teacher’s equity goals and impact on students.  

Furthermore, because justification facilitation is complex work and suggests potential for 

advancing equity for students, practice-based methods could be supportive for teachers to learn 

the practice and in ways relevant to their contexts and goals. During Josie’s exit interview, I 

asked, “Is there anything else you think would be helpful for researchers and/or teachers to know 

related to the role of justifying in the math class or how to provide students these opportunities?” 

Josie said watching examples of how some teachers facilitate justification opportunities can be 

helpful. She referenced a two-week long professional development that she attended two 

summers prior to the study that was particularly helpful for her, where she observed a teacher 

facilitating mathematical reasoning practices with students in relational ways. Josie 

recommended, “Just watch somebody because a lot of the things, a lot of these like micro 

opportunities really happen so fast, it would be impossible to like write it in a textbook, you 

know.” According to Josie, part of providing justification opportunities involves micro-level 

enactment moves that may not be included in all curriculum materials. She also mentioned that 

getting better at facilitating justification opportunities takes practice. Josie explained,  

I think it's like anything, it's like part of it is seeing it in practice, and part of it is practicing. And 

the more you do it, the more of those small moments you'll recognize and know how to inject a 

justification opportunity into. So you have to do it a lot and you have to see somebody effectively 

do it to recognize it. You know, it's exposure, I guess. 

Learning to provide justification opportunities involves seeing examples of 

implementation in practice and practicing implementing it oneself. It takes time and practice to 

learn any new practice, particularly as it needs to be adapted to be culturally relevant and 

responsive to one’s own students, classroom context, and goals. Given the complexity of 

justification facilitation, this suggests providing practice-based opportunities may be helpful in 

supporting teachers in learning to facilitate justification opportunities in ways that are responsive 

to students and situated in critical dispositions for equity. Practice-based opportunities might 

involve seeing examples of how other teachers facilitate justification opportunities in similar and 

  45 

 
different ways within their unique contexts (i.e., representations of practice) and engaging in 

approximations of practice to practice implementing some justification opportunities in mediated 

settings before trying out on real students to minimize students being “practiced on” (Grossman 

et al., 2009). Mediated practice opportunities, such as trying out the delivery of parts of 

justification opportunities, can give the professional development facilitator opportunities to 

offer feedback to a teacher and for the teacher to retry and refine justification-oriented 

implementation strategies (such as discourse moves for eliciting justification). While there may 

be concern that practicing in a removed setting could lead to prescriptive instructional methods 

across contexts, research has found that practice-based opportunities can support teachers in 

feeling more prepared later on when they go to implement new teaching practices with their 

students, and research has shown teachers are able to apply those practices they learned in ways 

responsive to their students (Kavanagh et al., 2020; Schiera, 2021).  

In the final question of the exit interview, I asked Josie, “Anything else you want 

researchers or teachers to know?” and she responded, “Everyone should do it! Kids really like it! 

They're really good at explaining themselves.” According to Josie, justification opportunities can 

be leveraged to intrigue students’ curiosity, enhance their engagement with and enjoyment of 

mathematics, contribute to the development of a positive mathematical identity, and support their 

agency in the learning process. This study illustrated that justification has the potential to be a 

powerful, equity-oriented mathematical practice a teacher can leverage to provide students the 

agency to make/defend choices and to generate mathematical meaning/discoveries, whereby 

students are taking ownership of the mathematics, reflecting and cultivating positive 

mathematical identities in the learning process regarding what it means to know, do, and be good 

at mathematics.  

  46 

 
  
 
 
 
 
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  50 

 
 
CHAPTER 3: THE ROLE OF JUSTIFICATION-ORIENTED PROBLEM-BASED 

LEARNING EXPERIENCES IN SUPPORTING STUDENTS’ AGENCY AND 

MATHEMATICAL IDENTITY 

Abstract 

Mathematical reasoning practices, such as justification, can advance equity for students 

by facilitating their agency in the learning process and supporting positive mathematical identity 

development in students. Problem-based learning lessons can provide a helpful context for 

teachers to leverage justification opportunities in ways that can advance these aspects of equity. 

This study examined how one Teacher of Color in a racially/ethnically diverse fourth-grade 

classroom of students in an urban, Title I school facilitated problem-based learning experiences 

that leveraged justification opportunities to support equity for her students. This article discusses 

how and why justification-oriented problem-based learning opportunities can be conducive for 

supporting a teacher’s equity goals. Curricular tasks, instructional strategies, and classroom 

vignettes from one teacher’s classroom are shared.  

Background 

The Standards for Mathematical Practice (SMP) specify the importance of students 

engaging in mathematical reasoning practices, such as reason abstractly and quantitatively (SMP 

1) and construct viable arguments and critique the reasoning of others (SMP 3) (National 

Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, 

p. 6). Mathematical reasoning practices engage students in sensemaking, promoting critical 

thinking skills and conceptual understanding of content (Bieda et al., 2022). When embedded in 

inquiry-oriented learning experiences (such as problem-based learning lessons), mathematical 

reasoning practices can advance students’ agency by providing them opportunities to explore and 

make mathematical discoveries themselves (Hansen, 2022; Schettino, 2016). In the problem-

based approach, “students are expected to solve problems or make sense of mathematical 

situations for which no well-defined routines or procedures exist. In introductory activities, as 

well as in application settings, students are expected to explore problems, make conjectures, and 

draw generalizations about mathematics concepts and processes” (Erickson, 1999, p. 516). In 

problem-based learning experiences, students are making mathematical discoveries, rather than 

having the mathematics directly imparted to them. Thus, the role of the teacher is to use students’ 

curiosity and ideas to drive learning. In problem-based learning, because students are making 

  51 

mathematical discoveries, this creates opportunities for students to generate and defend claims, 

or engage in the process of mathematical justification. Justification refers to “the process of 

supporting your mathematical claims and choices when solving problems or explaining why your 

claim or answer makes sense” (Bieda & Staples, 2020, p. 103). By design, problem-based 

learning experiences can provide opportunities for students to engage in mathematical 

justification. Justification is described by Bieda and Staples (2020) as an “equity practice” that 

can develop students’ agency. Specifically, Bieda and Staples (2020) state that opportunities to 

justify one’s reasoning supports students’ agency because students are generating mathematical 

ideas themselves in ways that make sense to them rather than practicing doing the teacher’s 

mathematics. There is limited research on both the justification facilitation process (Karunakaran 

& Levin, 2022) and on the relationship between justification and equity (Bieda et al., 2023). I 

argue that examining the process of how justification opportunities are facilitated by a teacher 

can help highlight ways in which justification opportunities are provided and how those 

opportunities relate to a teacher’s equity goals. Given how infrequent justification occurs in 

classrooms (Melhuish et al., 2020), it would make sense that a teacher who views justification as 

supporting their equity goals would incorporate the practice as part of their pedagogy. This study 

explored two questions: (1) What are one teacher’s goals for promoting equity in the classroom? 

(2) How does the teacher leverage the practice of mathematical justification to meet her equity 

goals? Understanding how justification opportunities are facilitated in the context of a teacher’s 

equity goals may help teacher educators better be able to support teachers in incorporating 

justification opportunities into instruction in ways that those teachers view responsive to their 

contexts, students, and goals.  

Conceptual Framework 

Framing this study is Schettino’s (2016) conceptualization of relational problem-based 

learning, which considers the interaction between concepts of justification, curriculum, 

ownership of knowledge, and shared authority. I also draw on Berry’s (2020) conceptualization 

of agency and identity to frame how these constructs are dimensions of equity and how they are 

interrelated. Then, I discuss the role of distributing intellectual authority (Zavala & Aguirre, 

2024) in facilitating justification opportunities in ways that can empower students’ agency and 

build positive mathematical identities. 

  52 

 
Relational Problem-based Learning with a Justification Focus. Problem-based learning is a 

student-centered pedagogy and relational learning process that involves mathematical reasoning 

practices, including justification (Erickson, 1999; Schettino, 2016). In describing the “essential 

elements” of the problem-based approach, Erickson (1999) states, “students can explore and 

make conjectures, listen to one another’s explanations and justifications, and reach consensus 

about the concept or about how particular ideas are related” (p. 518). Because students are tasked 

with making discoveries themselves rather than being told by the teacher, this process involves 

students defending their claims and learning from ideas their peers have raised. Erickson 

explains, “In this teaching approach, students make reasoned conjectures about a problem-

solving task, justify their thinking, and listen to and consider other students’ ideas” (Erickson, 

1999, p. 518).” In addition to the problem-based approach affording opportunities for 

justification, I argue that justification is an integral component of problem-based learning.  

While there are many definitions and conceptualizations of problem-based learning 

(Merritt et al., 2017), I draw from Schettino’s (2016) framework of teaching mathematics with 

relational problem-based learning. Schettino describes problem-based learning as a relational 

and collaborative, not individual, process. In this framework, the themes that characterize 

relational problem-based learning are (1) ownership of knowledge, (2) justification- not 

prescription, (3) the connected curriculum, and (4) shared authority. Ownership of knowledge 

refers to centering students’ solution strategies, ideas, perspectives, and discoveries in the 

learning process. This requires that the students justify their claims and decisions, as the teacher 

is not prescribing methods for solving the problem. Scaffolding and supporting students in not 

having to rely on the teacher for solving problems with mathematical concepts that are very new 

to them requires a curriculum that coherently connects mathematical concepts to one another and 

also connects to students’ background knowledge. Sharing authority of knowledge means all 

members of the class are contributing to the knowledge building of the classroom, requiring 

teachers to decenter themselves. Schettino found that relational problem-based learning allows 

students to have more agency in the learning process, supports students in viewing mathematics 

more positively, and thus contributes to students seeing themselves as more confident doers of 

mathematics. 

Erickson (1999) says that facilitating problem-based learning requires the teacher to 

establish a classroom environment where students feel safe taking risks, such as disagreeing with 

  53 

a peer, and comfortable sharing ideas that are still uncertain or in-progress, without fear of 

embarrassment or ridicule. Schettino (2016) also echoes the importance of a comfortable 

classroom environment for facilitating problem-based learning, particularly because it is such a 

relational process. Schettino (2016) states, “In the context of creating a relational learning 

environment, empowering student voice and agency is facilitated by creating a safe environment, 

further demonstrating the interdependence of the relational framework on each of its parts” 

(Voice and Agency section, para. 3). According to Schettino, a safe environment is an essential 

part of supporting students’ agency and voice through problem-based learning. 

Agency and Identity as Equity Constructs. Advancing equity for students involves supporting 

their development of a positive mathematical identity and facilitating their agency in the learning 

process. Berry (2020) elaborates the National Council of Teachers of Mathematics’ (NCTM’s) 

Principles to Actions and frames equity using the interrelated constructs of identity and agency. 

According to Berry, teaching mathematics in ways that advance equity for students involves 

more than just teaching mathematical skills but also empowering students to see themselves as 

capable, confident doers of mathematics (identity). This involves integrating students’ cultural 

experiences and contexts into the mathematics as well as expanding students’ perceptions of 

what it means to do and be good at mathematics (which extends beyond quick calculations and 

correct answers, to also include engaging in mathematical discourse, collaboration, and 

productive struggle). According to Berry, when students develop a positive mathematical 

identity, this in turn can empower their agency to make decisions that increase their participation 

and engagement in mathematics class. Berry (2020) states, “If one identifies themselves as being 

smart and good at mathematics, then they present themselves and adopt behaviors and actions of 

smartness and being good at mathematics” (slide 16). This may include more frequently asking 

questions, participating in mathematical discourse, and internalizing increased motivation to 

persevere. Citing Aguirre and colleagues (2013), Berry states, “Agency is our identity in action 

and the presentation of our identity to the world” (slide 15). Having agency within the 

mathematics classroom and beyond is a reflection of having a positive mathematical identity. 

Echoing the importance of cultivating students’ identity and agency in mathematics, in the 

November 2024 NCTM president message, Knighten (2024) cites the NCTM Catalyzing Change 

series which states, “Mathematics instruction should be consistent with research-informed and 

equitable teaching practices that nurture children’s positive mathematical identities and strong 

  54 

sense of agency.” Knighten explains that connecting to students’ cultural experiences, building 

from what they know, and providing opportunities for students to learn from and with their peers 

allows students to “take ownership of the learning process” and “put themselves in the driver’s 

seat.”  

Figure 8 shows the relationship I conceptualize between relational problem-based 

learning and equity. Problem-based learning (with a focus on relational components that include 

justification) can advance equity for students through supporting their development of a positive 

mathematical identity. Having a positive math identity can, in turn, facilitate students’ agency. 

As students have more agency in the learning process, they are also building a more positive 

mathematical identity. As a result, the constructs of agency and identity can positively influence 

one another. 

Figure 8. Framework for Relational Problem-based Learning Facilitating Students’ Agency and 
Positive Mathematical Identity Development 

Distributing Intellectual Authority to Support Students’ Agency and Identity Development. 

Supporting students in building positive mathematics identities and empowering their agency in 

the learning process involves the teacher sharing authority of knowledge with students. Zavala 

and Aguirre (2024) state that distributing intellectual authority can “make space for multiple 

forms of knowledge and communication” (p. 98). They explain that moving from a teacher-

centered to student-centered mathematics classroom is one way to advance students’ 

mathematical authority. Zavala and Aguirre (2024) define distributing intellectual authority as 

involving, “positioning students as mathematical authorities in the classroom, using participation 

structures that expand student mathematical contribution opportunities, and embracing the role of 

teacher as facilitator and co-constructor of knowledge in the classroom” (p. 99). They explain 

that this involves providing multiple opportunities for students to communicate what they know 

and have choice in how they do so, and these opportunities must be frequent enough and 

  55 

 
 
facilitated in ways that allow students to see themselves and each other as mathematical 

authorities. Supporting students in seeing themselves as mathematical authorities is part of 

building positive mathematical identities, and having mathematical authority inherently is 

advancing agency in the learning process. Zavala and Aguirre (2024) describe three 

“instructional approaches” involved in distributing intellectual authority: (1) position students as 

holders of math knowledge, (2) expand use of collaborative participation structures, and (3) 

embrace the teacher-as-facilitator role. Regarding the first, positioning students as holders of 

math knowledge means recognizing the multiple forms of background knowledge, experiences, 

and perspectives students bring, and eliciting and leveraging these to provide space and support 

for students to apply their ideas and strategies to analyze new mathematical situations in which 

they can generate their own ideas and justify them using the mathematics. The second 

instructional approach involves providing students frequent and varied opportunities for 

collaboration. Collaborative participation structures Zavala and Aguirre (2024) name as 

conducive to distributing intellectual authority include “turn and talk,” “think-pair-share,” 

“collaborative group work,” and “gallery walks.” These all provide varied ways for students to 

voice their ideas, learn from other’s thinking, expand their curiosity by asking questions, and 

build upon others’ ideas. The third instructional approach involves the teacher taking on a role as 

facilitator and co-constructor of knowledge, rather than the traditional math teacher role of 

imparting all knowledge. This means providing space for students to engage in productive 

struggle and utilize questioning strategies that engage students in generating and justifying 

mathematical claims and ideas they notice, discover, and wonder about. Zavala and Aguirre 

(2024) affirm, “When intellectual authority is distributed, more voices, ideas, and actions can be 

leveraged for individual and collective mathematics learning” (p. 97). Distributing intellectual 

authority is a key component involved in facilitating justification opportunities, particularly in 

ways that aim to advance students’ agency and build positive mathematical identities. 

Sampling and Participant Background 

The participant in this study is one teacher, Josie (her chosen pseudonym). Josie was 

chosen for the study through a combination of convenience sampling methodology due to 

proximity and my pre-existing relationship with her as a former colleague and purposeful 

sampling methodology based on existing teaching practices and equity-oriented dispositions 

(Merriam & Tisdell, 2015). When considering participants for the study and speaking with Josie 

  56 

 
about it, she mentioned that she starts every math lesson with a Number Talk (Parrish, 2014) and 

that she incorporates justification opportunities within and outside of Number Talks. Because 

Josie knew me as a teacher too at one point, this aided in providing a foundation for a trusting, 

researcher-participant relationship during the study. 

Josie had nine years of teaching experience at the time of the study, with the most recent 

six years being at the school in which the study took place. The school is a Title I, K8 public 

charter school, located in an urban district in the Midwest with students predominantly from 

historically and currently marginalized backgrounds, including Black and Latinx students. 

During the study, Josie taught in a self-contained, fourth-grade classroom. Most of her classroom 

experience was in elementary classrooms, situated in urban communities with students of 

racially/ethnically diverse backgrounds and in low socioeconomic regions. She also had some 

experience as a middle school mathematics teacher. Josie identifies as a Filipina woman, and she 

immigrated to the United States from the Philippines when she was 13. From once working 

together with Josie at the same school, I knew that she had (and the school’s mission was about) 

strong commitments to social justice and place-based learning.  

Data Collection and Analysis 

Study Design. The research questions guiding this study focused on how Josie provided 

justification opportunities in her classroom to meet her equity goals. The first question this study 

explored was, What are one teacher’s goals for promoting equity in the classroom? The second 

was, How does the teacher leverage the practice of mathematical justification to meet her equity 

goals? These questions were examined through case study methodology (Merriam & Tisdell, 

2015). In this study, Josie’s practice and perspectives (about the relationship between equity and 

justification) were the “case” in which the study was bound.  

Data Collection and Analysis. To examine Josie’s practice and perspectives around equity and 

justification, data included recordings and transcripts of interviews and lesson planning 

meetings, field notes of classroom observations, and curriculum materials. Data collection began 

with a pre-observation in November 2023 to get a general idea of Josie’s practice and inform 

some pre-interview questions, such as clarifying questions about types of justification 

opportunities observed. In January 2024, a two-part pre-interview (totaling 130 minutes) was 

conducted to gather background information (e.g., Josie’s identity and teaching experience), 

  57 

 
understand Josie’s equity goals, and gather her general perspectives on the relationship between 

justification and equity.  

Classroom observations were conducted two days per week (typically consecutive) from 

mid-February through April 2024, totaling 16 observations. Observing lessons during the second 

half of the year (and before standardized testing) provided a helpful timeframe for examining 

lessons with classroom routines well-solidified into the school year by Josie, and well-

established prior to my entry into the classroom space. These observations spanned beginning, 

middle, and end of units, and across topics including multiplying and dividing by powers of ten, 

division with remainders, foundational fraction concepts (e.g., meaning), creating equivalent 

fractions, adding/subtracting fractions, and early decimals concepts (e.g., naming, meaning). 

During classroom observations, field notes were taken on Josie’s justification-oriented, problem-

based enactment strategies and on student justifications and agency. These field notes included 

types of student justifications (e.g., justifications of their ideas, claims, choices, or mathematical 

relationships) and Josie’s justification implementation (including norms and pedagogical 

strategies for eliciting justification, such as discourse moves). Notes were also taken on instances 

in which Josie provided opportunities for supporting students’ mathematical identity 

development or agency (such as what she would say to decenter herself as the mathematical 

authority) and instances of or statements made by students exhibiting agency, such as 

questioning a peer.  

Spread throughout these observations were six (30-60 minute) lesson planning meetings, 

where Josie debriefed the previous week’s lessons and shared plans for the upcoming week. 

Specifically, Josie debriefed the past week’s justification opportunities, which included reflecting 

on questions I asked about observed lessons in relation to justification and her equity goals. For 

instance, I asked Josie what successes and challenges she experienced in supporting justification 

in the past week. I also asked Josie to reflect on what she noticed about students’ agency and 

identity in the past week, and how she was considering these noticings in her plans for the 

upcoming week. Sometimes I asked questions to clarify or better understand something I 

observed in a lesson, about what happened during a day I could not observe that week, or about 

lessons earlier in the school year. For instance, after observing a Number Talk that connected to 

their class’ reading challenge, I asked Josie if she had done any Number Talks earlier in the 

  58 

school year that related to something happening in class or in the school, and to tell me about 

those types of Number Talks and why they were important to her. 

During the observation window, Josie shared all curriculum materials she implemented 

into a folder in Google Drive. These included tasks I observed and did not observe so that I could 

get (1) a sense of curricular coherence and (2) a better understanding of how her lessons built 

upon one another, potentially in ways that supported students’ agency during different stages of a 

unit. Analysis was conducted on all tasks in this folder. Keyword analysis was used to identify 

words and phrases within the curriculum materials related to justification, such as why, prove, 

true, explain, pattern. Thematic analysis (Braun & Clarke, 2006) was used to look for themes in 

tasks with potential connections for supporting students’ agency or identity. One example of a 

theme that emerged was task design for cultural relevance. A task was grouped into the theme of 

cultural relevance when it related to students’ experiences, interests, or community (e.g., a task 

that referenced the amount of rainfall in the students’ city), because of the connection to their 

community. Another theme centered around patterns. An example of a task that was grouped into 

the theme of patterns included the direction, “Describe a pattern that you notice is true about 

multiplication for numbers that end in zeros.” This theme connected to agency because students 

were being tasked with ownership of knowledge by developing a conjecture about what happens 

when numbers are multiplied by powers of ten, rather than the teacher telling them.  

In May, a 60-minute exit interview was conducted to gather the teacher’s final reflections 

about justification and equity concepts, ask questions that spoke across observations or emerged 

from the study, and gather the teacher’s advice for other teachers looking to advance students’ 

agency and identity through justification practices.  

Analysis of pre- and exit interviews, lesson planning meetings, and field notes of 

classroom observations were also conducted using a thematic analysis approach, to look for 

connections between Josie’s equity goals and her justification-oriented curricular enactment. One 

example of a theme that emerged was the construct of curiosity. Josie talked about inspiring 

students’ curiosity in interviews and in lesson planning meetings. Grouped into this theme 

included transcript quotes that explicitly used the word, curiosity. For example, the quote, “I just 

think if a child or person is more curious than afraid, they're more willing to try,” illustrates the 

teacher’s desire to support students in developing positive mathematical identity. Additionally, 

being curious is connected to facilitating students’ agency to engage with the mathematics as it 

  59 

provides a starting point to build from, which is to uncover or better understand something 

stemming from what one is curious about. Another theme centered around background 

knowledge. Grouped into this theme included references to background knowledge that the 

teacher mentioned in interviews or lesson debrief meetings, such as this quote from the exit 

interview, “All kids need the background knowledge to enter into the problem-based space, like 

the thinking of the problem, I guess. They need to be able to use their background knowledge 

and their reasoning as they justify their thinking.” Field notes were also analyzed to look for 

statements or questions the teacher asked to connect to students’ background knowledge. For 

example, in observation 14, the teacher directed students to think about their answer in a 

previous problem and consider what they know about money and fractions to answer the next 

question, “What fraction of a dollar is 5 dimes?” The exact statement said by the teacher that was 

grouped into the theme of activating background knowledge was, “Then I asked what fraction of 

a dollar is your answer to question 5… You know what money is, you know what a fraction 

looks like.” Activating students’ background knowledge was an important theme because of the 

connection to the knowledge students bring (their identities) and how drawing on background 

knowledge can support students’ agency in the learning process by using what they know to help 

themselves get started without the teacher’s help. Themes that emerged from thematic analysis 

showed connections to Zavala and Aguirre’s (2024) framing of distributing intellectual authority. 

In the findings, I describe Josie’s equity goals as situated within the three instructional 

approaches for distributing intellectual authority. Then, I share classroom examples that showed 

how Josie facilitated relational problem-based learning with a justification focus, as connected to 

her equity goals.  

Findings 

For Josie, advancing equity in mathematics required intentional focus on facilitating 

students’ agency and supporting students in building positive mathematical identity, which she 

saw as intertwined. This involved engaging students in problem-based learning experiences that 

focused on noticing and justifying patterns, rather than imparting these herself, and decentering 

herself as the mathematical authority. When examining the justification opportunities that Josie 

provided, problem-based learning emerged as a prominent theme. Discussed below are Josie’s 

perspectives on student agency and identity in mathematics instruction, which show connections 

to Zavala and Aguirre’s (2024) framing of distributing intellectual authority. Then, two cases are 

  60 

shared of how Josie facilitated relational problem-based learning experiences (focused on 

justification) to support her equity goals. All names presented in the findings are pseudonyms. In 

quotations where the word “like” was interjected as a filler and not needed for context, it was 

sometimes removed for readability. 

Josie’s Equity Goals: Supporting Agency and Identity. Josie saw agency and identity as 

intertwined, and relational problem-based learning experiences with a justification focus as 

inherently cultivating both of these. How Josie thought about the relationship between 

justification, agency, and identity has connections to Zavala and Aguirre’s (2024) 

conceptualization of distributing intellectual authority. According to Zavala and Aguirre (2024), 

distributing intellectual authority provides opportunities for collaboration and for students’ 

voices, ideas, and reasonings to be heard, and doing this requires the teacher to share authority of 

knowledge with and among students. Zavala and Aguirre (2024) describe three instructional 

approaches for distributing intellectual authority: (1) position students as holders of math 

knowledge, (2) expand use of collaborative participation structures, and (3) embrace the 

teacher-as-facilitator role. This section describes Josie’s perspective on the relationship between 

justification, agency, and identity as situated within Zavala and Aguirre’s (2024) framework of 

three instructional approaches for distributing intellectual authority. 

Position Students as Holders of Math Knowledge: Josie provided justification opportunities for 

students because it was important to her that her students saw themselves as holders of math 

knowledge, and she leveraged justification opportunities to center students’ reasoning as the 

mathematical authorities. Supporting students in generating and justifying ideas for themselves 

involved not answering many questions students asked that would stop them from thinking or 

would give away answers. Josie shared, “I always try and promote that I'm not the knower of 

everything… I just don't answer questions… So they know if I'm not answering a question, 

they're gonna stop asking because I want them to work through it.” Not answering or imparting 

the knowledge sought by students’ questions (and instead, supporting students in producing the 

knowledge and answers themselves) was Josie’s way of showing that she is not the “knower of 

everything.” Josie used similar language when she talked to me about not wanting to be the 

“keeper of math.” She shared, “I very rarely try and give the answers to them, ‘cause, you know, 

I think I've said this before, but I don't want to be the keeper of math. I want them to be the 

thinkers.” 

  61 

Josie supported students in being the thinkers by providing justification opportunities that 

provided students entry points into the problem, activated their background knowledge, and 

inspired their curiosity. She explained, 

Justifying allows everyone to have, to contribute, to have an entry point. So even if you can't do 

long division, you can explain it with a diagram by cutting, you know, tick marks into whatever. It 

allows everyone to share ‘cause everyone came to the classroom with a tool belt of knowledge. 

And they can share whatever is in there, and we can add to it and everyone can contribute, so I 

think that is part of agency. 

According to Josie, all students are holders of math knowledge because they all “came to 

the classroom with a tool belt of knowledge,” and they can learn from and build off of one 

another’s ideas to work toward collective understanding. In another conversation, Josie added 

that sometimes students need to be reminded about or supported in accessing the knowledge they 

bring, and inspiring their curiosity is one way that can activate that. Josie stated, “When I teach 

math, I want kids to be curious and know that they, if they really looked at it, they know more 

than they’re realizing.” 

Expand Use of Collaborative Participation Structures: Because Josie viewed all students as 

holders of math knowledge, she created frequent opportunities for students to collaborate so they 

could learn from one another. Josie explained that with twenty-four students in the room, there 

are so many perspectives and ideas to be learned from, outside of her own. She shared, 

It's common that I’ll allow them to talk to each other, too, which is really important, like give 

them time to either be with a partner or a small group. And be very clear that everybody should 

contribute, you know. I think that's important, too, especially for those tasks that are looking at a 

lot of numbers and really trying to notice as much as they can. The more brains you have, 

obviously the more you might notice, and more conclusions might come to something. 

In another conversation, Josie echoed these ideas, sharing that all students come to the 

classroom as different people, who think and learn in different ways, and one students’ way of 

unique thinking could be what helps something click for another student, especially with there 

being twenty-four students in the class. She explained, “They come to your classroom as 

different people… You never know what's really going to help them click in their head. So if 

there's twenty-four different people sharing a perspective, there's more chances that that might 

happen.” 

  62 

 
Embrace the Teacher-as-Facilitator Role: Because supporting students’ agency and building 

positive mathematical identities were prominent equity goals for Josie, she saw her role as that of 

a facilitator. For Josie, being a facilitator involved leveraging justification opportunities, situated 

within relational problem-based learning experiences, that centered students’ ideas and 

reasonings. When asked about the relationship she saw between justification and agency, Josie 

rhetorically asked,  

Don't they kind of go hand-in-hand? Or to me, it feels that way… The opportunities to have their 

thinking be heard and valued and uplifted is kind of the agency part…I think that justification 

opportunities and conversations inherently provide kids with agency.  

According to Josie, the mathematical practice of justification and agency “go hand-in-

hand.” In particular, Josie described justification as affording opportunities for students’ ideas 

and perspectives to be “heard and valued and uplifted,” (an expression and reflection of students’ 

identity in the learning process), which inherently advanced agency. Further describing how 

justification opportunities provided spaces for students’ voices to be heard, reflecting their 

identities and facilitating their agency as they access they mathematics, Josie explained, 

The reasoning is your own. You and your voice is, you know what I mean? You're able to voice 

your own critical thinking of whatever it is. So nobody told you this is a solution or this is a 

strategy. You did the math yourself, so that part is like identity… You do the task without I guess 

being trapped in an algorithm. You don't have to follow the step-by-step process right now. You 

can, you know, think of it in your own way. I think that allows for a lot of access. 

Being a facilitator meant not telling students a solution or strategy to use and letting them 

apply their own way of solving the problem. Facilitating students’ motivation and confidence to 

do so involved inspiring their curiosity. For Josie, reasoning and curiosity were directly related. 

She believed, “How could a child be curious if they're not allowed to ask questions and explore 

those questions? That's, you know, reasoning in a classroom.” When asked about how 

justification supported her goals as a teacher, Josie shared, “Math isn't just like you show up, you 

do the thing, and you get the answer. You're allowed to be more creative with it. And I think a lot 

of kids do well with creative tasks, because they can kind of put themselves into it.” Facilitating 

students’ creativity and joy for learning was important for Josie, because to her, math was more 

than calculating answers. Josie mentioned, “I love learning. I love engaging a curious mind, 

whether it's a kid or an adult, and I couldn't imagine doing any other job that would be 

fulfilling.” Josie went on to add, “There's so much joy coming from being curious, you know– 

  63 

the desire to learn, the desire to find out about something you just decided to be interested in 

because of whatever reason. I don't know what would happen to somebody that lost that.” For 

Josie, being curious was more conducive to students’ mathematical identity development and 

agency building than having to know answers right away, and helping them see this sometime 

involved boosting their confidence. Josie mentioned, 

If you understood that you could be curious and ask questions and explore something instead of 

having to know it right away. But then you, there's a lot of like self-confidence that has to be built 

with a kid that sees math and problems that way. 

Providing students opportunities to be curious, creative, and discover and justify 

mathematical ideas themselves through problem-based learning meant taking on a role as a 

facilitator, which meant leveraging justification opportunities. For Josie, justification 

opportunities provided the space for her students’ ideas to be heard, backed up by the 

mathematics to support their claims and ideas. Being able to approach problems using their own 

knowledge, in turn, facilitated their agency in the learning process. The teacher-as-a-facilitator 

role was important to Josie so that they could be the thinkers. Josie believed, “Because really, 

that's where we're leading them to is like to be outside of our walls and be free thinking humans 

out in the world.” For Josie, building students’ positive math identity (through creative 

opportunities to engage in reasoning) can support students in being agentic, free thinking humans 

within and beyond the mathematics classroom. 

Discovering and Justifying Patterns. To support these goals of students being free-thinking 

humans, Josie provided students opportunities to discover and justify mathematical patterns and 

relationships themselves through problem-based learning experiences (in ways that reflected 

their identity), rather than her imparting these ideas to students (facilitating their agency). These 

opportunities were leveraged at the beginning of new topics and units, by connecting to students’ 

background knowledge. When directly asked about the relationship she sees between problem-

based learning, justification, and background knowledge, Josie said, “All kids need the 

background knowledge to enter into the problem-based space, like the thinking of the problem, I 

guess. They need to be able to use their background knowledge and their reasoning as they 

justify their thinking.” Problem-based learning involves engaging students in noticing patterns, 

activating background knowledge to apply in the investigation of those patterns, and justifying a 

rule or reason that can explain the mathematics behind those patterns. During the justification 

  64 

process, those patterns become understood and then ultimately proven or disproven. For Josie, 

math was all about noticing patterns, and this was why she created opportunities for students to 

discover and justify patterns at the beginning of new units. Connecting again to her theme of 

“gamifying” math, Josie explained, 

So much of math is patterns, looking at patterns, thinking about patterns, or understanding why 

those patterns happen. And I think that is a way to kind of ‘gamify’ the experience for kids. And 

they're good at that… And especially at the beginning of units, if you didn't tell them yet what 

something is, they can just look at the raw numbers, or whatever it is, and see those things and be 

like, ‘Hey, actually, I came up with this.’  

For Josie, math discussions were fundamental parts of the pattern-noticing and 

justification process, and all of this was important for cultivating their agency and building 

positive math identity. Josie shared, 

I love, you know, doing math discussions, and I think that engages the kids, too, to feel like they 

can share ideas, you know, and not just like, “Hey Miss. [Josie] told me to do this specific thing 

for this problem.” But they could be like, “I noticed this pattern. What if we tried this pattern 

across these other things, will we see the same thing?” And then they ask like, “Well, why didn't 

we…? Or why did we…?” So we always start with discussion, and that's built into every day for 

us, too. 

Described below are two cases from Josie’s class where she facilitated students’ agency 

by engaging them in discovering and justifying patterns in their own ways that supported 

building their mathematical identities. The curricular tasks, Josie’s enactment strategies, and the 

students’ ideas are highlighted. In both of these cases, students noticed patterns, developed a 

rule, and engaged in a justification process aiming to prove or disprove it. 

Figure 9. Continuum of Noticing to Justifying Patterns for Proving/Disproving Math Rules 

Case 1: Investigating Divisibility Rules:  During two consecutive lessons, the class was 

investigating divisibility patterns of single digit numbers. Although Josie could have provided 

students with a list of “divisibility rules,” supporting her equity goals of advancing students’ 

agency and building positive mathematical identity meant providing students a problem-based 

learning experience for discovering and justifying these divisibility rules or patterns themselves. 

Prior to this lesson, Josie gave students a pre-assessment on reasoning with divisibility to gather 

  65 

 
 
some of their background knowledge before implementing this task. For Josie, building from and 

connecting to background knowledge is a key part of facilitating problem-based lessons. Josie 

created her own worksheet for students so that they could have an organized space where they 

could record their ideas as they explored patterns in numbers divisible by particular digits. The 

directions on the worksheet said, “Find the pattern or make up a rule for divisibility by single 

digit numbers.” The worksheet had two columns. Running down the left column was a row for 

each digit 1-10 (with 7 omitted). There was a box with each of these digits that left space for 

students to show their work for the patterns they noticed with numbers divisible by each of these 

digits. The specific worksheet direction in that column said, “What patterns do you notice? Show 

your work.” Running along the right-hand column was space to record, “Our Group’s Rule.” 

This was a two-day lesson, where students spent the majority of class investigating divisibility 

patterns/rules with a partner, and then gathering as a whole class at the end of the lesson to 

discuss what they noticed that day. Students investigated these patterns by choosing their own 

numbers to divide and see what they noticed, and skip counting to see what they noticed. Josie 

recorded the rules students noticed each day on an anchor chart (see Figure 10). When discussing 

a rule for divisibility by 1, the conversation was short, as was the justification, but it appeared 

sufficient for Josie as she did not spend much time on it:  

Student: Every number is divisible by 1. 

Ms. Josie: Can you prove that? 

Student: With division you cut into groups. I can cut any number into groups of 1. 

The discussion that came about during numbers divisible by 3 was more complex. Below 

is a segment from one pattern students noticed, highlighting Josie’s questioning strategies for 

supporting their justifications. (The chart in Figure 10 shows other patterns students also noticed 

and justified with numbers divisible by 3). 

Ms. Josie: What patterns and rules did you notice for numbers I can divide by 3? 

Student 1: It goes odd, even, odd, even. 

Ms. Josie: Is that only true for 3s or is it also true for other numbers? 

Student 1: It also works for... (and other students jumped in to add more examples). 

Ms. Josie: So, because it’s an odd number, it alternates. Why does it do that? 

Student 2: I think it’s because 3 is made up of an odd number and an even number. 

Ms. Josie: Tell me how. 

Student 2: 3 is a 2 and a 1. 

  66 

Ms. Josie: Is every odd number made up of an odd and an even part? Think about it for a minute. 

63, 77. 

Here, Josie asked students if this pattern is only specific to 3s or if dividing by other 

digits presents this alternating pattern too, to help students think about the nature of 3 versus 

other numbers. Josie also asked why the numbers alternate in the way that they do. When a 

student said that 3 is made up of an odd and even part, Josie pressed for reasoning by asking 

how, and she presented some examples to consider.  

While Josie could have presented all the divisibility rules for students upfront to “save” 

instructional time, this would not have advanced her equity goals of facilitating students’ agency 

and allowing students’ identities to be reflected in the learning process. Allowing students to 

discover divisibility patterns and justify the rules they came up with themselves in ways that 

made sense to them, promoted agency and afforded students the opportunity to build positive 

mathematical identity, and also supported students in a deeper exploration and understanding of 

numbers and division. 

Figure 10. Chart of Students’ Divisibility Noticings 

  67 

 
 
When reflecting on this lesson in a debrief meeting later, Josie said she felt, that across all 

the lessons they did in the past week, these two days provided students the most opportunities to 

engage most deeply in the process of mathematical justification. For her, that meant, “time to 

actually sit and play with numbers.” When I asked how Josie thought this lesson supported 

students’ agency or mathematical identities, she talked about the task being accessible because 

anyone can notice a pattern and share their ideas, whether correct or incorrect. Linking how 

access supports agency, Josie explained, 

Anybody can notice a pattern. And everybody feels just like they can participate, you know. Like, 

‘I can look at this and share a thought, and it's not gonna be wrong. I'm just looking for a pattern 

to share out loud, and whatever it is fine, as long as we have proof that it's true, based on, you 

know, the numbers that we see. So I guess, I don't know, like agency, just anyone can start, at 

least start it, even if they don't get to as much depth as like–  there's varying degrees of, you 

know, math that the kids get to. But everyone can start with something. 

Elaborating on students’ agency, Josie talked about the importance of providing students 

time to notice, explore, and explain. 

I think, you know, I often feel rushed for time. But I think it's important to give them the time to 

even just sit, even if they're not productive, you know, in that 15 minutes the whole entire time, or 

for that 20 minutes. Just to be able to sit there and look at it and putz around a little bit and do 

something else. And then come back to and be like, “Oh, actually, now that I'm sitting here, I 

notice this.” I often feel like, you know, impatient, like okay, let's just get to work. But I think 

sometimes I have to remember just let them look at it. Because I mean, most of these kids are 

curious learners anyway. So they want to, they want to be able to share something. They love 

sharing their thinking, so I think just giving them time to sit and really get to find something out 

on their own. 

Case 2: “1 is equal to 10 in the decimal world”: In a later lesson that was connecting the end of 

a unit focused on fractions to a new unit focused on decimals (and the relationship between the 

two), Josie gave students the task, “Write the fraction ½ as a decimal.” At this point in time, 

students had not yet learned about writing fractions as decimals in school. In order to engage 

students in this task, Josie activated students’ background knowledge and supported them in 

noticing patterns. This exploration led to an interesting discussion around place value and Zoe’s 

(all names pseudonyms) metaphor of “the decimal world.” 

  68 

 
As a bridging question leading up to this task, the prior question students were asked to 

answer and justify was what fraction of a dollar is 50 cents. Students answered with 50/100, 

5/10, and ½. The class agreed that one dollar represents 100 cents. One student’s justification 

was, “Because 1 is half of 2, and 50 is half of 100. If the top number is half of the bottom 

number, it’s equivalent to ½.” Another student added, “So we all know 5/10 is ½. Let’s say we 

multiply both those by 10. That’s 50/100.” Noticing that the numerator was half of the 

denominator, and checking by multiplying a numerator and denominator by the same number, 

convinced the class that 50 cents can be represented as fractions that include 50/100, 5/10, and 

½.  

This background knowledge that was activated engaged students in noticing a particular 

pattern that could be extended to the next question of considering how ½ would be written as a 

decimal. During this task, Josie pulled out a blank number line as a tool to support one student’s 

process of justifying her claim that ½ is written as 0.5. Josie guided the class in filling out the 

number line to support them in noticing any place-value related patterns in how decimals are 

written as they relate to fractions. As Josie guided students in filling in the number line with their 

noticings, she scaffolded by asking students to look at the patterns. Investigating these patterns 

was pivotal to the justification process of showing ½ is 0.5, and ultimately extending students’ 

understanding of place value, fractions, and decimals (and how they relate). Figure 11 shows 

how the students thought through the place value relationship of whole numbers, fractions, and 

decimals to create the decimal number line, as recorded on the projector by Josie. 

Zoe: Zero point five. 

Ms. Josie: Why? 

Zoe: One whole is like a ten, in the decimal world. It’s kind of hard to explain. 

Ms. Josie: It’s okay, you’re doing a good job. 

Zoe: On the number line, if you went–  

Ms. Josie: I have a number line right here. I knew you were going to say something about the 

number line, so I had one handy. 

Zoe: 1 is equal to 10 in the decimal world. If you cut the 1 in half, it’s not going to be five. 

Ms. Josie: Like this? 

Zoe: Kind of. 

Ms. Josie: These represent tenths. Zoe is kind of getting there. 1 is 10, in this example of the 

decimal world. 

  69 

Ms. Josie: What is 2? 

Zoe: Two tenths. 

Ms. Josie: How would I write two tenths? 

Zoe: 2/10 

Ms. Josie: If each one, if the unit fraction is the 10, what would I call this? 

Zoe: Four tenths. 

Ms. Josie: Could I fill out the rest with this pattern? 

Multiple Students: Yes. 

Ms. Josie: If I was here at the dot, what would this fraction be called? Look at the pattern. 

Multiple Students: Six tenths. 

Ms. Josie: What about this one? 

Multiple Students: Seven tenths. 

Ms. Josie: I’m looking for the fraction, three tenths, as a decimal. Who could come up to the 

board and find it? Even though they’re written as a fraction or decimal, their word form is the 

same. This [points to 4/10] is four tenths and this [points to 0.4] is four tenths. What is this 

fraction? 

Multiple Students: One tenth. 

Ms. Josie: What’s this fraction? 

Multiple Students: Zero tenths. 

Ms. Josie: What’s this decimal’s name? What fraction is it equivalent or equal to? 

Multiple Students: One half. 

Ms. Josie: 5/10 or ½ 

Ms. Josie: [And] this one? 

Multiple Students: One whole. 

Ms. Josie: Ten tenths or one whole.  

Figure 11. Number Line 

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Josie took out a decimal number line to help support the class in understanding Zoe’s 

claim that ½ is written as 0.5 as a decimal. When Zoe presented this claim, she did not yet have a 

justification developed and said it was “kind of hard to explain.” Though Zoe did have a partial 

justification, which was “one is equal to ten in the decimal world,” she was still trying to figure 

out how to describe this with mathematical clarity and evidence. Because Zoe mentioned a 

number line, Josie took out a number line to help elaborate Zoe’s claim in a way that started with 

the ideas Zoe initiated. Though Josie seemed to lead much of the justification development, she 

used Zoe’s claim (i.e., ½ is written as 0.5 as a decimal) and Zoe’s starting justification for this 

(i.e., “one is equal to ten in the decimal world”) as building blocks to scaffold the class in 

connecting the relationship between whole numbers, fractions, and decimals together on a 

number line, which helped illustrate Zoe’s idea that “one is equal to ten in the decimal world,” 

and ultimately produce justification for the claim, “½ is written as 0.5 as a decimal.” 

In debriefing this lesson later, Josie talked about why math discussions are important to 

have in class and how a student’s idea can sometimes convey a concept to other students in a 

way she would not have thought of. Referring to Zoe, Josie explained, “Honestly, she helped me. 

She was like, she said it in a way that I never would have said it. And I was like just trying to 

repeat it in my own words, to see if like, if I said it again, who would see it again?” The design 

and enactment of this problem-based lesson afforded Zoe the opportunity to explain her own 

working understanding of decimals and fractions through a metaphor that made sense to her, 

reflecting a positive mathematical identity. Josie supported Zoe’s agency in the learning process 

by working to understand her metaphor, helping elaborate Zoe’s idea through a process of 

explanation and justification, and supporting the class in understanding Zoe’s metaphor. Josie 

explained that having math discussions is important because every class is different and “they 

come to your classroom as different people.” Cultivating students’ mathematical identity means 

recognizing that each student brings to class different perspectives and providing opportunities 

that give students the agency to apply their perspectives to the content in ways that make sense to 

them. Sometimes one student’s idea can help other students. Josie affirmed,  

I think that's why I think these conversations are so important and discussion is so important. 

‘Cause you really don’t know. Every group is different. Their context is different. Their 

perspective is different. They come to your classroom as different people. And you can plan all 

you want, but you never know what's really going to help them click in their head. So if there's 

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twenty-four different people sharing a perspective, there's more chances that that might happen. I 

don't know what, how that helped the kids that nodded, but it did. It helped them click something. 

The design and enactment of Josie’s problem-based lessons, in which she leveraged the 

mathematical practice of justification, supported her goals of facilitating students’ agency and 

providing opportunities for students’ identities to be reflected in the learning process. 

Foundation of Conducive Classroom Environment. For Josie, supporting students’ agency 

and positive mathematical identity development in problem-based lessons required a conducive 

classroom environment. Cultivating a conducive classroom environment was a culture built over 

time through development of norms, trust, and relationship-building with students over the 

course of that school year. How her students viewed what it meant to do and be good at 

mathematics came from the class culture established from the start of the year and how that 

culture became reinforced each day into the year. Klosterman (2016) characterizes a conducive 

classroom environment as a classroom culture where students feel safe taking risks and sharing 

their in-progress ideas. As Josie’s students were working to justify their claims, they took risks 

by sharing ideas they were not yet certain of but tried to back-up with mathematics. For Josie, 

setting the foundation of a conducive classroom environment meant talking through “math 

agreements” with students at the beginning of the school year and creating them over the course 

of several days. Josie shared, 

We spent a whole week making the agreements. So we were like, ‘Everyone is going to feel like 

they contribute to this classroom and you're gonna be a person that helps in that.’ You know what 

I mean? Not prevents people from their own learning. And I think they understood that. So, I 

think part of it was the kids buying in and willing to be those kind of people for each other... I 

think part of it is a culture thing from the school, but also, we– I just, I really pushed really hard 

to make it so that they knew I was serious about holding them to these agreements, you know? 

And yeah, we just talked a lot about everybody deserves a chance to learn, everybody can 

contribute. Everybody knows something, and you know what I mean? I think because we also had 

those discussions and they helped make the norms and they were the ones that dictated what my 

job was as a teacher.  

Figure 12 shows the math agreements the class came up with. The student agreements 

started with ideas that Josie presented to the students that they built from. The teacher 

agreements were made by the students. The green posters show the teacher agreements, and the 

yellow posters show the student agreements. Josie reinforced these throughout the year, evolving 

  72 

the agreements into norms. Students learned that in their mathematics classroom, everyone’s 

ideas were valued, mistakes were part of the learning process, authority of knowledge was shared 

amongst, and that they had a responsibility to understand one another and hold one another 

accountable. 

Figure 12. Math Agreements Developed by Ms. Josie’s Class 

When I asked Josie which of the agreements she thought might be most supporting 

students’ agency, she said that they all do. Josie explained,  

They all in a way ask the students to be critical thinkers and be able to share that critical 

thinking. So not just be students that can, say, memorize what I said and then do it. But that they 

have the agency to solve things, to think about things, to focus on what somebody else is saying 

which holds the, you know, the speaker accountable to their peers, not just to me. 

For Josie, it was important that students saw themselves as doers of mathematics and that 

they had opportunities to construct knowledge themselves, which is why she incorporated 

problem-based approaches to mathematics pedagogy. Josie shared,  

I think a student that sees themselves as capable they’re more willing to share their thinking… I 

think that that is more important than me being an authority of knowledge in a classroom, and as 

  73 

 
 
 
part of learning, because I cannot give my knowledge to somebody. I can’t. That's never– you 

know what I mean? If I could, my job would be so easy. 

Josie cultivated a conducive classroom environment so strong that students even felt 

comfortable challenging their peers whom they consider “math whizzes.” Josie’s definition of 

“math whiz” was a student who can see patterns very naturally and who often produces correct 

answers. In describing this dynamic in her classroom, Josie elaborated, 

I know that the kids perceive Jonah especially, even since second, third grade, because he's been 

in Faith’s (Josie’s daughter) classes like the math whiz or whatever. He's just really natural with 

it. He understands patterns really well. And so I know the kids kind of see that. But even 

sometimes, and I don't want them to always go with what they say, you know. And sometimes they 

don't, which I'm like, okay so they're still thinking for themselves, and they perceive not Jonah as 

the keeper of math, but all of them as able to do math, and they can disagree with even somebody 

that, like Jonah, who is often right. 

For Josie, it was important that students did not see her or the “math whizzes” as the 

“keepers of math,” but rather, all see themselves as knowers and doers of mathematics. Josie 

mentioned that often the math whizzes will still share ideas that are wrong. 

‘Cause sometimes in math, what's intuitive isn't always, you know, the answer or something, and 

sometimes you didn't–  Evan will get something that is– seems intuitive– like it should be this, and 

it's not. And so there's been instances where that's happened. 

Jonah and Evan were students that Josie specifically referred to as math whizzes. A major 

testament to the conducive classroom environment Josie created can be seen relative to the 

identities of the class’ math whizzes. For one, the students in her class felt comfortable 

countering or disagreeing with Jonah and Evan because they could also see themselves as 

knowers and doers of mathematics. Second, the identities of Jonah and Evan were not centered 

around having to always be right (which can sometimes happen with kids used to excelling in 

mathematics class), as shown by their willingness to present ideas that were in-progress and 

potentially be mathematically incorrect.  

Discussion and Conclusion 

This study examined how one teacher provided justification opportunities to meet her 

equity goals through mathematics instruction. Specifically, the study explored the questions, 

What are one teacher’s goals for promoting equity in the classroom? and How does the teacher 

leverage the practice of mathematical justification to meet her equity goals? Josie’s goals for 

  74 

 
advancing equity for students meant facilitating their agency in the mathematics learning process 

and supporting students’ development of positive mathematics identities. This involved 

relational problem-based learning experiences that included inspiring students’ curiosity, sharing 

authority of knowledge with and among students, supporting students in discovering and 

justifying mathematical patterns and relationships themselves, and establishing a conducive 

classroom environment by which they felt comfortable and empowered to engage in these 

opportunities. 

Given the limited research in the field on how the process of justification opportunities 

are facilitated (Karunakaran & Levin, 2022), how infrequent justification occurs in classrooms 

(Melhuish et al., 2020), and the gaps in the literature on the role of justification in advancing 

equity (Bieda et al., 2023), this study helped highlight examples of how justification could be 

leveraged to support a teacher’s equity goals. Because Josie’s equity goals included supporting 

students’ agency and positive mathematical identity development, this suggests that justification 

may be a powerful practice in advancing critical equity dimensions. Importantly, understanding 

how justification is facilitated in the context of a teacher’s equity goals could better help teacher 

educators support other teachers in incorporating justification into their instruction in ways that 

are relevant to each teacher’s context and goals. Because every classroom context is different, 

and practices are contextually situated, teacher educators need to be able to work within a 

teacher’s equity goals in order for teachers to view the practice as valuable and to be leveraged in 

ways conducive to their students. 

It can be easy to wonder whether Josie’s practice and perspectives are about equity or if 

her approach to mathematics instruction is simply “just good teaching.” Often, equity and good 

teaching are terms used interchangeably. If good teaching, by definition, means instruction that 

is supporting positive learning experiences and producing positive student outcomes for all 

students (particularly marginalized students), it would seem that good teaching would inherently 

advance aspects of equity. Equity is a broad word and multidimensional construct that signifies 

positive learning experiences and an impact of positive outcomes for all students. If equity 

means positive student outcomes are realized (especially for marginalized students), then good 

teaching practice is a means by which those can become achieved. This paper defined Josie’s 

primary equity goals as a desire to empower students’ agency and advance their mathematics 

identity development, and the study explored a teaching practice by which Josie strove to 

  75 

 
achieve those equity goals. For Josie, teaching practice that incorporated relational problem-

based learning approaches (that leveraged justification opportunities) was the type of “good 

teaching” she perceived to advance equity for her students. In the article, “But That’s Just Good 

Teaching! The Case for Culturally Relevant Pedagogy,” Ladson-Billings (1995a) affirms that 

culturally relevant pedagogy is indeed a description of good teaching, stating, 

One usual response to what I share is the comment around which I have based this article, “But, 

that’s just good teaching!” Instead of some “magic bullet” or intricate formula and steps for 

instruction, some members of my audience are shocked to hear what seems to them like some 

rather routine teaching strategies that are a part of good teaching. My response is to affirm that, 

indeed, I am describing good teaching, and to question why so little of it seems to be occurring in 

the classrooms populated by African American students” (p. 159).   

In following work, Ladson-Billings (1995b) studied the pedagogical practices of eight 

teachers in classrooms with Black students and found that the “good teaching,” which was 

achieved through culturally relevant pedagogy, rest on three criteria: (1) students achieve 

academic success (2) students develop cultural competence (3) students develop a critical 

consciousness through which they challenge inequities. It makes sense that pedagogy that is 

relevant to students’ lives, where students see content connected to their culture and community 

and where they have a voice to analyze and critique that content, can support access (to a better 

understanding of the content) and engagement (through finding meaning in the content). For 

Josie, who taught in a classroom of predominantly Students of Color, it was important to her that 

she connected mathematics content to students’ lives, that students saw themselves as doers of 

mathematics and as a community (a mathematical community and community more broadly as 

affiliated with their shared place), and that students had choice and voice in the learning process.  

Equity is a complex and multifaceted construct. The broad use of terms such as equity 

and culturally relevant pedagogy in the literature has implications for the field to be clearer 

about the ways they are defining and using these constructs. Gutiérrez’s (2009) framework for 

dimensions of equity is one example of a helpful framework that names and describes the 

multiple facets of equity, within the context of mathematics teacher education. Gutiérrez defines 

equity as comprising four interrelated dimensions (access, achievement, identity, and power), all 

of which are described as needed to realize equity. Josie believed that justification can promote 

all of these equity dimensions. When I asked, “Where do you see the practice of justifying within 

each of these dimensions?” Josie stated, “Justifying supports all of those things,” and went on to 

  76 

  
elaborate how. Although Josie saw justification as a practice that can promote equity (because it 

can promote all four dimensions), the focus on this paper was on her attention to the critical 

dimensions in order to illuminate the relationship between justification, agency, and identity. 

When the field discusses equity goals, it is important to be intentional about attending to aspects 

of critical dimensions of equity to broaden our understanding beyond the dominant dimensions 

of equity, as access and achievement have dominated the discourse, according to Gutiérrez. 

Since Gutiérrez also states that all dimensions must work together to realize equity, more 

attention to the critical dimensions can lead the field closer to supporting equitable outcomes and 

experiences for all students. 

More research is needed to understand the relationship between justification opportunities 

embedded in relational-problem-based learning experiences and students’ agency and identity. In 

particular, how and why teachers provide these opportunities across various contexts needs 

further exploration. Because justification opens spaces to center students’ voices and ideas and 

requires the teacher to be responsive to the perspective brought by students, the field could 

benefit from research that explores the potential of culturally relevant and culturally responsive 

justification practices. In particular, what would pedagogy that facilitates opportunities for 

culturally responsive justification look like, and what impact could it have on equity for 

students? How might culturally responsive justification further advance students’ agency and 

positive mathematical identity development? This has implications for the development of 

frameworks or tools that seek to understand what culturally relevant and responsive justification 

could look like and how to support educators in facilitating this type of pedagogy. Additionally, 

while this study only measured the teacher’s perceptions of equitable experiences, future studies 

could benefit from a deeper analysis of the student experience. For example, understanding 

students’ own perspectives on the effects that culturally relevant or responsive justification 

opportunities have on their agency and identity would be important for understanding the 

alignment between the teacher’s intentions and the impact on students. This could involve 

surveying students about their perspectives on how they feel about their own level of 

understanding and how they view what it means to do and be good at mathematics (including 

how they situate themselves within that description), throughout the year. A better understanding 

of how and why teachers can facilitate justification opportunities to meet their equity goals (and 

  77 

 
the impact on students) could better help teacher educators support teachers in leveraging 

justification in ways that are responsive to teachers’ contexts. 

  78 

 
 
 
REFERENCES 

Berry (2020). Using identity and agency to frame access and equity. National Council of  

Teachers of Mathematics. 

Bieda, K., Conner, A., Kosko, K. W., & Staples, M. (Eds.). (2022). Conceptions and  

consequences of mathematical argumentation, justification, and proof. Springer. 

Bieda, K., Staples, M., Doherty, K., Matthews-Johnson, A., & Hwang, J. (2023). Considering the  

alignment between teachers’ conceptions of justification and their visions for equitable 
instruction. Research report in Proceedings of the forty-fifth annual conference of the 
North American Chapter of the International Group for the Psychology of Mathematics 
Education, vol. 1, 124-130. Reno, NV. 

Bieda, K. & Staples, M. (2020). Justification as an equity practice. Mathematics Teacher:  

Learning and Teaching PreK-12, 113(2), 102–108. 
https://doi.org/10.5951/MTLT.2019.0148   

Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative research in  

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Mathematics Instruction. The Mathematics Teacher, 92(6), 516-521. 
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Gutiérrez, R. (2009). Framing equity: Helping students “play the game” and “change the game.”   

Teaching for Excellence and Equity in Mathematics, 1(1), 4-8. 

Hansen, E. K. S. (2022). Students’ agency, creative reasoning, and collaboration in mathematical  

problem solving. Mathematics Education Research Journal, 34(4), 813-834. 
https://doi.org/10.1007/s13394-021-00365-y 

Karunakaran, S. S., & Levin, M. (2022). Justification in the Context of Tertiary Mathematics:  

Undergraduate Students Exploring the Properties and Relations of the Dihedral Group. In 
Conceptions and Consequences of Mathematical Argumentation, Justification, and Proof 
(pp. 239-250). Springer, Cham. 

Klosterman, P. (2016). Identification and establishment of social and sociomathematical norms  
associated with mathematically productive discourse. Doctoral dissertation, Washington 
State University. 

Knighten, L. (November 2024). Positioning students as learners and doers of mathematics.  

National Council of Teachers of Mathematics, President’s Messages. 
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President/Archive/Latrenda-Knighten/Positioning-Students-as-Learners-and-Doers-of-
Mathematics/  

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Ladson‐Billings, G. (1995a). But that's just good teaching! The case for culturally relevant  

pedagogy. Theory into practice, 34(3), 159-165. 
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Ladson-Billings, G. (1995b). Toward a theory of culturally relevant pedagogy. American  

educational research journal, 32(3), 465-491. 
https://doi.org/10.3102/00028312032003465 

Melhuish, K., Thanheiser, E., & Guyot, L. (2020). Elementary school teachers’ noticing of  
essential mathematical reasoning forms: Justification and generalization. Journal of 
Mathematics Teacher Education, 23(1), 35-67. https://doi.org/10.1007/s10857-018-9408-
4 

Merriam, S. B., & Tisdell, E. J. (2015). Qualitative research: A guide to design and  

implementation. John Wiley & Sons. 

Merritt, J., Lee, M. Y., Rillero, P., & Kinach, B. M. (2017). Problem-based learning in K–8  
mathematics and science education: A literature review. Interdisciplinary Journal of 
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National Council of Teachers of Mathematics (NCTM). (2014). Principles to actions: Ensuring  

mathematical success for all. Reston, VA. 

National Governors Association Center for Best Practices & Council of Chief State School  
Officers. (2010). Common core state standards for mathematical practice. 
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Parrish, S. (2014). Number talks: Whole number computation. Sausalito, California: Math  

Solutions.  

Schettino, C. (2016). A framework for problem-based learning: Teaching mathematics with a  
relational problem-based pedagogy. Interdisciplinary Journal of Problem-Based 
Learning, 10(2), 12. https://doi.org/10.7771/1541-5015.1602 

Zavala, M. R., & Aguirre, J. M. (2024). Cultivating mathematical hearts: Culturally responsive  

mathematics teaching in elementary classrooms. Corwin Press.  

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CHAPTER 4: INCORPORATING MATHEMATICAL JUSTIFICATION AT THE 

BEGINNING OF UNITS TO SUPPORT STUDENTS’ ENGAGEMENT WITH NEW 

CONTENT AND AGENCY IN THE LEARNING PROCESS 

A fourth-grade class just began their first unit on fractions, having not yet had any lessons 

on fractions this school year. The teacher introduced the task shown in Figure 13. As expected, 

the most common student answers shared were ⅓ and ½. One student’s justification for the 

answer of ⅓ was that 1 out of 3 pieces are shaded in. One student’s justification for the answer of 

½ was that the full bottom half of the shape is shaded in. Class that day ended on a cliffhanger, 

having not come to a consensus on the answer to this problem. 

Figure 13. First Fractions Problem 

In response to students’ answers and justifications for their thinking, the teacher prepared 

a hands-on lesson for engaging students in an exploration and discussion of thirds, halves, and 

the meaning of fractions the following day. The teacher gave students the direction, “Use this 

Play Doh to make a pastry, and cut your pastry into thirds.” In groups of two or three, students 

used their Play Doh to form giant cookies, brownies, marshmallows, and other desserts. Then 

  81 

 
 
 
they cut their pastries into three pieces, some of equal sizes and some of different sizes. The 

teacher asked students to justify their reasoning for cutting their pastries in the ways that they 

did. She asked one group who made a cookie that they cut into three equal-sized pieces, “Why 

did your group choose to cut your cookie this way?” After eliciting their reasoning, she asked the 

class about one group’s marshmallow cut into three pieces of different sizes, “What do you think 

about this cut?” The students’ justifications paved the way for a productive discussion on the 

meaning of fractions, which will be elaborated in this paper. Using this context, this article 

discusses how the mathematical practice of justification can be leveraged at the beginning of a 

new unit in ways that support students’ engagement with new content and agency in the learning 

process. 

Establishing a Foundation for Fraction Concepts 

Having a strong conceptual understanding of fractions is important for setting a 

foundation for the building blocks of higher level math concepts later, such as proportional 

reasoning and algebra. In fourth grade particularly, where students will learn to create equivalent 

fractions and add and subtract fractions with like denominators (that are more complex than the 

fractions they worked with in third grade), it is foundational to first understand the meaning of 

fractions as being equal-sized parts of the same whole. A teacher may find it first helpful to 

activate students’ background knowledge and check for their understanding about the meaning of 

fractions. Activating students’ background knowledge, in ways that invite opportunities to 

consider notions of justice related to equal shares, alongside their recollection of fraction 

concepts, can be generative for supporting engagement. Children often become invested in topics 

around justice, fairness, and equal shares, which can hook them into the content and potentially 

support them in recalling what they may have learned about fractions in the past. Keeping in 

mind that fairness does not mean sameness, the word “fair” as used in the classroom dialogue 

shown later (and as used thereon out), will be used to refer to “equal shares” (as opposed to 

sameness). Having opportunities to explore and discuss fraction concepts encourages students to 

really think about fractions, understand that fractions have meanings, and notice that the pieces 

(of unit fractions which share the same whole) should be equal. Leveraging the mathematical 

practice of justification can help provide these meaning-making opportunities for students.  

  82 

The Role of Mathematical Justification 

Bieda and Staples (2020) define mathematical justification as, “the process of supporting 

your mathematical claims and choices when solving problems or explaining why your claim or 

answer makes sense” (Bieda & Staples, 2020, p. 103). Justification is a mathematical practice 

that can benefit both students’ learning and teachers’ instruction. For students, the justification 

process can support students in talking about mathematics and promote knowledge mobility 

(Liljedahl, 2020), increase engagement, strengthen reasoning skills, support conceptual 

understanding of content, and foster positive mathematical identities and productive dispositions 

towards mathematics (Boaler, 2022; Staples et al., 2012). For teachers, eliciting students’ 

justifications can help with assessing students’ working understanding of content at a given time 

(Staples et al., 2012). Eliciting students’ ideas about fractions, scaffolding their understanding 

about fraction concepts, and supporting them in the communication of and reasonings behind 

their ideas involves the teacher decentering themself as the mathematical authority. Sharing in 

authority of knowledge with students opens up the space for students to have the agency to be 

able to pose ideas, generate their own claims, and justify the claims or ideas they come up with 

and decisions they make as they are exploring new content at the beginning of a unit. Bieda and 

Staples (2020) describe justification as an “equity practice” because it can support both students’ 

access to content and their agency in the learning process. This article builds from Bieda and 

Staples’ conceptualization to consider how the beginning of a unit can be particularly conducive 

for leveraging mathematical justification to facilitate students’ agency and engagement with new 

content. 

Classroom Discussion of Fractions 

The dialogue below features snippets of the discussion facilitated by Ms. Josie (all names 

pseudonyms) in which she engaged students in discussion about their reasonings and 

justifications for the ways they cut their Play Doh pastries. Ms. Josie leveraged the justification 

process in ways that supported students’ engagement with the content in the new unit and their 

agency during the learning process. This involved activating students’ background knowledge, 

eliciting justifications of students’ ideas, and connecting to the theme of “fairness” (i.e., equal 

shares in this case) to scaffold students’ understanding of fractions as equal-sized pieces. Ms. 

Josie’s students felt comfortable sharing their uncertain or in-progress ideas because of the 

  83 

 
comfortable class environment Ms. Josie cultivated over time and the norms around math 

discussion participation established from the start of the school year. 

Discussion of Cookie Cut into Three Equal-sized Pieces. 

Ms. Josie: Why did your group choose to cut your cookie this way? (See Figure 14). 

Malik: The reason my group cut it that way is because it was the easiest way to cut it instead of 

finding the most complex way. 

Ms. Josie: Why is it the easiest? 

Malik: It’s the easiest way, than trying to find a harder way. It was easier to make them equal. 

Ms. Josie: Why did you care about making it equal? 

Malik: We didn’t want to find a more complex way.   

Figure 14. Play Doh Cookie Attempted to Be Cut into Three Equal-sized Pieces 

In the dialogue above, Ms. Josie pressed for students’ reasoning in their justification for 

cutting the cookie in the way that they did. She did this by asking Malik why his group chose to 

cut his cookie in that way, why it was the easiest way for them, and why they cared about making 

it equal. From this conversation, all we know is that the shape in which the group partitioned the 

cookie was the easiest way for them to cut it into thirds, but the concept of fractions having 

equal-sized parts not did not clearly surface.  

Next, Ms. Josie moves on to elicit justifications from a group that cut their Play Doh 

pastry into pieces of unequal sizes. In the dialogue below, Ms. Josie decides to engage the class 

in discussion about the cut of one group’s marshmallow (not pictured) into different sized pieces 

to see what discussion that would generate regarding the size of parts. She begins by asking the 

class what they notice about the cut of the marshmallow, then asks students how they would feel 

about receiving different pieces of a dessert, and finally transitions to asking the class about the 

naming of the pieces. Ms. Josie uses the justification process to tie together students’ noticings of 

  84 

 
 
 
 
and feelings about portion sizes of dessert shares, with naming conventions and meaning of 

fractions.  

Discussion of Marshmallow Cut into Three Different-sized Pieces. 

Ms. Josie: What do you think about this cut? 

Shana: It looks uneven. 

Ms. Josie: How would you feel if you had this piece (points to the smallest piece) and the other 

two people had these pieces? 

Taylor: It would be unfair because that size piece is way smaller. 

Ms. Josie: But the other day, we said this is still one-third because it’s 1 of 3 pieces. So if I were 

sharing this cookie with you and I said I’m getting one-third and you two are also getting one-

third, even though these pieces are different sizes, can I say that? 

Multiple Students: It’s unfair.  

Jordan: You can take a piece off the bigger piece to give to the person with the smaller piece. 

Ms. Josie: Let’s say I went to the coffee shop with Amir. And I said, let’s split this cookie in half, 

(reusing the cookie from the previous group as a manipulative). What would you say if I said this 

piece (points to the bigger piece) is my half and the other is his half? 

Multiple Students: Unfair!  

Ms. Josie: Half is a fraction right? How would I write one-half? Write it with your finger in the 

air. 

Adriana: There’s a 1 on the top, then a line, then a 2 on the bottom. 

Ms. Josie: You’re saying these both are still written as ½? 

Jonah: No! That’s not a half! 

Destiny: That’s like one-quarter. 

Ms. Josie: So if I really did want to cut this cookie in half, where would I cut it? 

Destiny: Down the middle. (Comes to the projector and cuts the cookie down the middle). 

Destiny: Now that’s one-half. 

Ms. Josie: Why is this one-half now? 

Destiny: Because it went down the middle. 

Ms. Josie: This piece and that piece are the same... 

Multiple Students: Size!  

In the dialogue above, Ms. Josie appealed to the idea of fairness in equal shares of dessert 

portions and referenced back to a more familiar fraction (one-half) to activate students’ 

background knowledge and scaffold their understanding towards a justification of fractions 

  85 

 
having equal-sized pieces. After Shana described the three marshmallow pieces as “uneven,” Ms. 

Josie asked the class how they would feel if they received the smaller piece and others received 

the larger pieces, all while referring to each piece as “one-third.” Multiple students expressed it 

would be unfair, and Taylor’s justification for the reason it was unfair (and why multiple other 

students likely felt it to be unfair) was because the piece was smaller. Ms. Josie reminded the 

class, though, that on the previous day they had said that one-third is one of three pieces. Next, 

Ms. Josie proceeded to cut the first group’s cookie into two (distinctly different-sized) pieces 

(see Figure 15) and asked if that represented half, because it is two pieces. Students were familiar 

with the fraction ½ meaning two equal-sized pieces since the partitioning is “down the middle,” 

which was a justification that seemed to help activate background knowledge and scaffold 

students’ understanding that the fraction of thirds should also signify equal-sized pieces.  

Figure 15. Play Doh Cookie Cut into Two Pieces 

Summarizing the Discussion. 

Ms. Josie: What’s an important thing we need to know about fractions based on what we just 

did? 

Aaron: They have to be equal! 

Ms. Josie: What does? 

Mya: The pieces. 

Ms. Josie: The pieces that I cut have to be equal. 

Mya: That doesn’t look equal. 

Ms. Josie: You’re right. It’s hard for me to cut this exactly equal. 

Ms. Josie: If I cut something into fourths, all of the pieces have to be? 

Multiple Students: Equal!  

  86 

 
 
 
After engaging in discussion about halves meaning two equal-sized pieces and thirds 

meaning three equal-sized pieces, Ms. Josie did a quick check with students to ask about fourths 

and what would be important to remember about the size of fourths. In third grade, the students 

likely learned that fractions signify equal-sized pieces of a whole, but the hands-on activity Ms. 

Josie facilitated that elicited justifications (e.g., for their decisions on the cuts they made and 

their ideas about fractions and sharing) helped her elicit what students remembered and activate 

background knowledge about fractions in ways that the students cared about and made sense to 

them (e.g., fairness of having equal-sized shares of desserts). These justification opportunities 

supported students’ engagement with the content and their agency in the learning process. 

Debriefing on Students’ Agency and Engagement with New Content in the Lesson 

When debriefing this lesson, I asked Ms. Josie, “How do you think students engaging in 

that process of justifying supported their mathematical identities and agency?” Ms. Josie 

explained,  

I guess, maybe they just like, force them to really think about fractions, especially really common 

ones, like halves and thirds. And that those fractions have a real meaning, like not just 1 out of 

however many, 3 pieces, but that the pieces should be equal. Or, you know, you're going to get 

unfairness, or maybe a wrong recipe, or that the idea that the pieces of a fraction are equal was 

important ‘cause they felt it. They were mad about it. 

Curious to understand more about her pedagogical approach, I inquired, “If you just told 

them that fractions need equal size pieces, and then just moved on without letting them explore, 

what effect do you think that would have had?” Ms. Josie reflected on having done that in the 

past and shared,  

I've definitely done that. I think once we move on to equivalent fractions– because this has 

happened before– where, when the pieces aren't equal exactly. For example, if you have, you 

know, one-third and two-sixths or something. I think it's harder for kids to see that equivalence 

then if they don't really understand about, like, “Here's the whole, two of the same whole, and 

that here are the equal pieces that they are cut into.” So that's happened before. I think, just, I 

don't know, maybe [students] not like [having] as deep of an understanding, and just like 

[thinking], “These are fractions and I know there's a top number and a bottom number,” but 

never have to think about– Like, how in the green problem [students justifying], “Oh, it's 1 out of 

3 pieces, so that must be one third.” So having that still same misconception. 

  87 

 
From past experience, Ms. Josie learned that quickly reminding or telling students that 

fractions need to have equal-sized pieces does not provide a strong enough foundation for 

understanding the meaning of fractions needed for solidifying equivalent fraction concepts in 

ways that set them up for success later in the unit. Rather, through her years of teaching 

experience, Ms. Josie found that providing students justification opportunities actively engaged 

them in learning new content and supported their agency in the learning process. In the case of 

the beginning of this unit, justifications that connected to halves being cut down the middle and 

the notion of fairness in equal shares of dessert portions, supported students in understanding that 

thirds (and other unit fractions that share the same whole) have equal-sized pieces. Reflecting on 

how sharing authority of knowledge with students and providing students opportunities to 

explore and justify their ideas is important for both supporting students’ agency and engagement 

with new content, Ms. Josie shared, “I cannot like give my knowledge to somebody. Like I can’t. 

That's never– you know what I mean? If I could, my job would be so easy.” Although it takes 

more time than direct instruction to provide students opportunities to explore, pose and justify 

ideas in ways that build from their background knowledge, students are developing their agency 

in the learning process and engaging more actively with the content.  

Guidance for Your Classroom 

This section elaborates what Ms. Josie did in her classroom to leverage justification 

opportunities for supporting students’ agency and engagement with new content, and includes 

guidance for your classroom. Discussed are the affordances of the beginning of a unit, the role of 

sharing authority of knowledge with students, the design of exploratory tasks, and connections to 

formative assessment that supports a responsive approach to instruction. 

  88 

 
 
Figure 16. Leveraging Mathematical Justification to Facilitate Students’ Agency and 
Engagement with New Content and to Facilitate Teachers’ Formative Assessment Opportunities 
to Inform Responsive Instruction 

Affordances of the Beginning of a New Unit. The beginning of a new unit is an especially 

conducive place in the curriculum for leveraging the mathematical practice of justification in 

ways that can support students’ agency and engagement with new content. This is because 

justification can serve as a bridge that connects students’ background knowledge to the new 

content to be learned.  

Figure 17. Justification Process to Bridge Students’ Background Knowledge and New Content 

At the beginning of a new unit, a teacher has an important choice: tell students the new 

mathematics that they can memorize passively, or give students the agency to explore, discover, 

and justify concepts themselves as active doers of mathematics. The beginning of new units or 

  89 

 
 
 
 
 
 
 
topics are particularly conducive places for supporting students’ agency and engagement with the 

mathematics exactly because the content is so new to students. Leveraging the mathematical 

practice of justification (in ways that support students in building on background knowledge 

from previous units or experiences) can be a helpful tool in this process of discovery and 

exploration of new content. Given that teachers have a loaded curriculum with extensive content 

to cover, it can be tempting to rely on direct instruction instead, to perhaps quickly impart to 

students new mathematical procedures, relationships, or meanings. However, this is not as 

engaging or effective long-term, which could result in taking time to re-teach later. Ms. Josie 

explained, 

You're still gonna have to go back and reteach the rest that didn't memorize it. ‘Cause really, 

that's what that is. If there's not like– the most, majority of your class is not gonna be a bunch of 

memorizers. So you're gonna have to spend that time anyway. 

Spending extra time at the beginning of a new unit in intentional ways sets a strong 

foundation that can support the solidifying of key concepts for students early-on, ultimately 

setting up students for success through the progression of the unit. Providing students 

exploratory opportunities to engage in justification, with an eye to where in the unit you are 

leading them to, can help you make decisions about what to spend more/less time on throughout 

a unit. Ms. Josie affirmed, 

But as much as you can give those opportunities, you know. Because, the reality of teaching is 

that there's not a lot of time and that curriculum, they jam pack it with all kinds of stuff. You feel 

like you can't get through all of it. But I think that's also why I say look at the big picture, look at 

your end goal for that unit, and pick the most relevant things that you want to cover. Sometimes it 

all also helps to know where you are leading kids to. 

Share Authority of Knowledge with Students. Sharing authority of knowledge with students is 

important for facilitating students’ agency, supporting students in activating their background 

knowledge, and ultimately for setting the stage for an active (versus passive) learning process 

that engages students. Decentering oneself as the mathematical authority means not readily 

giving away reasonings, definitions, or answers and tasking students to develop their own 

justifications instead. For example, instead of telling her fourth-grade students that one-third (or 

any unit fraction) means equal-sized parts, Josie engaged the students in discussion about 

fairness in ways that activated their background knowledge about a familiar fraction, one-half. 

After talking through a real-world scenario with students where a cookie could be cut into two 

  90 

 
pieces (of equal then unequal sizes), instead of telling students which represented one-half, Josie 

pressed for students to offer the justification. Destiny reminded the class that the cookie cut into 

two equal-sized pieces represents one half, with the justification, “because it went down the 

middle,” (referencing the cut going down the middle). Speaking to her pedagogy of sharing 

authority of mathematical knowledge with students, Ms. Josie shared, “I don't want to be like the 

keeper of math. I want them to be the thinkers.” In order for students to generate answers and 

procedures or to discover mathematical relationships and concepts themselves, they need time to 

explore and to be supported in drawing on their background knowledge to help them make 

connections to the new content, and to develop justifications of the claims they make through 

this discovery process. Because the teacher is not prescribing procedures or imparting 

definitions, answers, or reasonings, leveraging the mathematical practice of justification can 

support students in using mathematics to back up the claims and ideas they are exploring. In 

explaining her role as the teacher in this process, Ms. Josie explained, “I think it's my job to 

listen and understand students’ thinking, and then like, decide what my next move will be to help 

them get where I need them to be.” Ms. Josie finds that eliciting students’ justifications supports 

her in assessing what students know at a given time, which can help her build from students’ 

working understanding as a bridge to guide them toward the new content in ways that make 

sense to them.  

Sharing authority of knowledge with students means intentionally providing students 

opportunities to justify their ideas. To support students in communicating their ideas and claims, 

Ms. Josie suggests providing sentence stems, particularly at the beginning of the year if students 

have minimal experience engaging in mathematical discussion in previous grade levels. Though 

Josie did not provide sentence stems in this lesson (which occurred in April), she found that 

overtime students did not need them anymore after enough practice. One sentence stem 

suggested by Ms. Josie is, “Here is a pattern I noticed and I think this pattern happens 

because…” Others might include: 

I think… because I remember (from last year, from earlier in the year)... 

● 
●  My claim is… My warrant/reasoning is… 
● 
● 
● 

I conjecture… because I know… 
I used this solution strategy because… 
I tried or tested… because… 

  91 

 
Exploratory Task. Exploratory tasks, like the task Ms. Josie designed, can be conducive for 

supporting students’ agency through justification opportunities. I define an exploratory task as 

one that is open-ended, accessible, and intriguing. Open-ended means there are multiple ways to 

approach the problem and students are not prescribed a method to use. For instance, Ms. Josie 

did not tell students how to cut their pastries. As students were forming their pastries and cuts, 

some asked her if the pieces needed to be the same size and all she told students was, “What do 

you think?” and “We will talk about it as a group.” Accessible means all students have 

background knowledge or experiences that they are supported in drawing from to enter into the 

problem space. Ms. Josie’s students knew that “thirds” relates to the number 3 and they were 

familiar with halves meaning two pieces that are equal-sized because the whole is cut down the 

middle. Intriguing means the task hooks students in, piques their curiosity, is relevant to 

students, students care about it, and/or students are interested or invested in finding the answer or 

solving the task at hand. In the case of this lesson, students cared a lot about fairness (i.e., 

sharing desserts where everyone gets the same size portion) which hooked them into the task, 

particularly when they were presented with a scenario that they perceived to be unjust or unfair. 

Ms. Josie mentioned, “The idea that the pieces of a fraction are equal was important ‘cause they 

felt it. They were mad about it.” 

Formative Assessment. Eliciting students’ mathematical justifications provides teachers a 

window into students’ current thinking, which can support the responsive instruction required for 

facilitating students’ agency in a lesson. The only way to elicit students’ working understanding 

of any mathematical concept is to provide them opportunities to share their thoughts and 

reasonings about it. Asking students to explain and defend their claims, ideas, and decisions 

(whether written, pictorially, or verbally) helps teachers understand students’ conceptions, and 

potentially, misconceptions. Ms. Josie picked one group that cut their pastry into three equal-

sized pieces and one group that cut their pastry into three different-sized pieces to elicit their 

current understanding and thus inform her next instructional decision. After Malik stated his 

reasoning for cutting the cookie into equal-sized pieces, Ms. Josie pressed for justification by 

asking, “Why did you care about making it equal?” It could be easy to make the assumption that 

the reason Malik and his partner cut their cookie into three equal-sized pieces was because they 

understood the definition of “thirds” as three equal-sized pieces. However, eliciting Malik’s 

justification showed that was not clear. A circle is not easy to cut into three pieces, which makes 

  92 

sense that the reasoning for his type of cut could have had to do more with ease of cutting than 

equal shares. This prompted Ms. Josie to ask the class their thoughts on the cut of the 

marshmallow. After Shana said, “It looks uneven,” Ms. Josie asked the class, “How would you 

feel…?” about receiving the smaller piece, alongside reminding them that the previous day they 

had said one-third is 1 of 3 pieces. In response to continued unclarity on the meaning of thirds 

(but also engagement around unfairness or injustice regarding unequal shares of desserts), Ms. 

Josie activated students’ background knowledge on halves as a fraction which means two equal-

sized parts, not two parts of any size. This supported scaffolding students’ understanding of all 

unit fractions of the same whole having equal-sized pieces, as supported by engagement in a 

justification process about a scenario that interested students.  

Conclusion 

The beginning of a unit can be an especially conducive place to leverage the 

mathematical practice of justification in ways that help teachers elicit and build from students’ 

background knowledge to support their engagement with new content and agency in the learning 

process. This involves the teacher sharing authority of knowledge with students, tasking students 

with generating and justifying claims they make through an exploratory and discussion-based 

learning process. While the beginnings of units are particularly conducive places for leveraging 

justification to support students’ understanding of new content, justification is a mathematical 

practice that can be incorporated into any lesson at any stage of a unit to facilitate students’ 

reasoning, agency, and engagement with content. According to Ms. Josie, “You can insert 

opportunities to justify anywhere… It could be the smallest moment. It could be, like, Why do 

you think that?…Do you agree? Why do you think that?” The mathematical practice of 

justification is more than just asking students why. Justification is a powerful practice aimed at 

facilitating students’ agency, strengthening students’ reasoning skills, centering their voices, and 

increasing their engagement with content.  

  93 

 
 
 
REFERENCES 

Bieda, K. & Staples, M. (2020). Justification as an equity practice. Mathematics Teacher:  

Learning and Teaching PreK-12, 113(2), 102–108. 
https://doi.org/10.5951/MTLT.2019.0148   

Boaler, Jo (2022) Mathematical Mindsets: Unleashing Students’ Potential through Creative  
Mathematics, Inspiring Messages and Innovative Teaching. Jossey-Bass; 2nd edition. 

Liljedahl, P. (2020). Building thinking classrooms in mathematics, grades K-12: 14 teaching  

practices for enhancing learning. Corwin Press. 

Staples, M. E., Bartlo, J., & Thanheiser, E. (2012). Justification as a teaching and learning  

practice: Its (potential) multifacted role in middle grades mathematics classrooms. The 
Journal of Mathematical Behavior, 31(4), 447-462. 
https://doi.org/10.1016/j.jmathb.2012.07.001  

  94 

 
 
 
 
 
 
CHAPTER 5: CONCLUSION 

This dissertation examined how one teacher, Josie, provided opportunities for her fourth-

grade students to engage in the mathematical reasoning practice of justification and how these 

opportunities supported her goals for equity in a racially/ethnically diverse classroom of 

students. Because agency emerged as a prominent equity goal for Josie, all three articles 

analyzed the role that mathematical justification opportunities can play in facilitating students’ 

agency in the learning process. In particular, justification opportunities allowed students to share 

authority of knowledge during lessons, opening spaces for them to discover mathematical 

relationships themselves, generate claims, and justify their ideas. All of this supported their 

engagement with content and development of positive mathematical identities. Sharing the 

findings of this dissertation study for three major audiences in the field of education (researchers, 

teacher educators, and practitioners) could be helpful in the ongoing challenge of bridging 

research and practice. Article One (Chapter 2) provided an account of the multifaceted ways in 

which Josie facilitated justification opportunities, to provide a more comprehensive examination 

of the various factors involved so that researchers could build from the framework in future 

studies, given the complexity of justification facilitation. The factors that this article discussed as 

influencing Josie’s facilitation of justification opportunities were purposefully designed 

curricular tasks, justification-oriented task enactment strategies, a conducive classroom 

environment, and a disposition for critical dimensions of equity (with a focus on agency and 

related equity constructs such as identity). Article Two (Chapter 3) and Article Three (Chapter 4) 

focused in on selected examples from Josie’s classroom in ways that could help support teacher 

educators and practitioners (respectively) in translating Josie’s strategies for facilitating 

justification to be applied in and adapted to the contexts of their own classrooms and goals. 

Article Two described the affordances of relational problem-based learning in leveraging 

justification, and Article Three discussed how and why the beginning of new units could be 

conducive places in the curriculum for facilitating justification. Relational problem-based 

learning experiences and the beginning of units can both offer conducive opportunities for 

supporting students’ agency via a justification process. This is because relational problem-based 

learning lessons, and particularly lessons at the beginning of units, are supportive for engaging 

students in activating background knowledge in ways that they can apply to their noticings of 

  95 

new content; from there, students can develop and reason through claims and conjectures about 

the content through the justification process.  

This study sought to address gaps that exist in the literature on both the justification 

process (Karunakaran & Levin, 2022), the lack of justification opportunities in classrooms 

(Melhuish et al., 2020), and on the role of justification in promoting a teacher’s equity goals 

(Bieda et al., 2023). Given that this study only examined one teacher’s practice and perspectives, 

more research is needed to understand how justification may be leveraged across a diverse set of 

classroom contexts and teachers’ equity goals for their own students. Researching how a practice 

gets implemented across contexts, and how practice-based experiences might support teachers in 

learning new practices, can be helpful for addressing the concern about practices being 

implemented in uniform or prescriptive ways across contexts (Philip et al., 2018). Aligned with 

situative perspectives of learning (Peressini et al., 2004), I view practices to be taken up and 

implemented by teachers in relation to their contexts and goals. I also argue that practice-based 

methods can support teachers in learning new practices in ways that can be relevant and 

responsive to their classrooms and equity goals for their students, especially a practice as 

complex as justification. Below, I offer suggestions for future research to continue to examine 

how justification opportunities can be leveraged to meet teachers’ equity goals, and how teachers 

may be supported in learning the practice of justification. 

Implications 

This study demonstrated a powerful connection between mathematical justification and 

equity. Specifically, it illuminated how the process of justification can be leveraged to provide 

opportunities to support students’ agency in the learning process, which is a critical equity 

construct. Supporting students’ agency in the learning process can increase students’ engagement 

with content and help cultivate students’ positive mathematical identities (which are additional, 

and intertwined, equity constructs associated with agency). Supporting equitable experiences and 

outcomes for all students involves attending to multiple dimensions of equity, because as 

Gutiérrez (2009) states, all dimensions are needed to realize equity. This may involve focused 

attention to the non-dominant dimensions (e.g., dimensions of identity and power) which have 

historically gotten less attention, such as considering how to build students’ agency and identity 

development through mathematics learning in order to also support their access to mathematics 

and achievement in the subject.  

  96 

This study has implications for more research to better understand how justification 

opportunities can be leveraged to advance teachers’ equity goals across diverse contexts. This 

research might begin with additional case studies across contexts unique from one another 

geographically, demographically, economically, and in terms of curriculum used, teaching 

approach, and district leadership. Understanding nuances by context can illuminate similarities 

and differences in how and why justification opportunities are facilitated, as well as help explain 

challenges and successes that are experienced alongside teachers’ respective equity goals. 

Because every classroom context is different, with varying student needs and teacher goals, 

understanding how justification opportunities can be adapted across contexts will be helpful in 

supporting teachers in learning to facilitate justification opportunities in ways they view relevant 

to their equity goals and responsive to their students. Findings from additional case studies could 

inform participatory action research (PAR) studies, where researchers apply what they learned 

from these case studies to work alongside teachers to build their capacity for facilitating 

justification opportunities. Findings from the case studies and PAR studies could inform the 

design and delivery of professional development geared towards (1) teachers to learn strategies 

for incorporating justification opportunities into their own classrooms, where facilitators could 

work with them to examine how to apply those strategies in ways responsive to their contexts 

and equity goals and (2) teacher educators looking to work alongside pre-service and in-service 

teachers as they learn to facilitate justification opportunities in culturally responsive ways. 

Additional Case Studies. Additional case studies that seek to understand how more teachers 

across various contexts (e.g., diverse student cultures, varying geographic and economic regions, 

different grade levels, students with disabilities, teaching approach, curriculum used, etc.) 

incorporate regular justification opportunities can help the field better understand ways that 

teachers use this practice to be responsive to their students and advance their equity goals. This 

could involve examining the design and adaptation of their curriculum materials, their lesson 

planning processes, their enactment strategies, their classroom environment, and their personal 

equity goals for students, to look for similarities and nuances in the facilitation of justification 

opportunities across contexts and goals. Because justification offers opportunities to center 

students’ perspectives and ideas (and for the teacher to be responsive to these), case studies that 

seek to understand what culturally responsive justification practices look like would be helpful 

  97 

for better understanding the role of justification in building students’ mathematical identities in 

ways where they can see themselves reflected in the mathematics. 

Exploring the student learning perspective in mathematics classes with regular 

justification opportunities is also important for understanding the impact of these opportunities 

(e.g., the alignment between the teacher’s intentions or goals and the student experience). For 

example, if a teacher incorporates justification opportunities with the equity goal of increasing 

students’ agency in the learning process, it would be helpful to know whether students report 

experiencing agency in their mathematics class. Surveying and interviewing students throughout 

the school year about what their mathematics classes have taught them regarding what it means 

to do and be good at mathematics (and where they situate themselves along those views) could 

provide insight into their perception of their own agency as mathematics learners.  

Participatory Action Research. As the field learns more about how teachers across contexts 

leverage justification to meet equity goals (e.g., through more case studies), participatory action 

research (PAR) studies could be beneficial for teachers who may not be incorporating 

justification regularly but want to learn more about how to implement these opportunities in 

ways that support their visions for equity and are culturally responsive to their contexts. PAR 

studies could allow teachers and researchers to learn from/with one another over the course of a 

school year. Teachers and researchers could partner together to analyze, create, and adapt the 

design of curricular tasks to be more conducive to justification opportunities in ways that are 

intended to meet their equity goals and in ways that are culturally responsive. The teachers and 

researchers might debrief on the successes and challenges the teachers experience with their 

curricular enactment strategies for facilitating justification opportunities. For example, if 

facilitating student agency is a goal, this could involve examining questioning strategies or 

discourse moves that may have advanced or hindered students’ agency during the justification 

process. Analyzing power dynamics among students, and between the teacher and students, 

could also provide insight into who is considered a mathematical authority and how that 

influences students’ individual and collective contributions during the justification process. As 

discussed above, studying students’ experiences alongside the teacher’s would also be conducive 

for understanding the alignment between the teacher’s goals and impact on students. This could 

involve conducting classroom observations, collecting student work, administering surveys to 

students, and facilitating interviews and focus groups with students.   

  98 

Professional Development. Findings from these case studies and PAR studies could inform 

professional development for teachers and teacher educators. For teachers, professional 

development could be designed around supporting teachers in learning strategies for 

incorporating justification opportunities into their own classroom. These strategies would be 

informed by the successes and challenges experienced by teachers as they facilitated justification 

opportunities during the case studies and PAR studies. Distinguishing between macro level 

strategies and micro level strategies for facilitating justification could also be helpful. Macro 

level strategies might include those for establishing a classroom environment conducive to 

justification, and micro level strategies could include particular discourse moves for eliciting 

students’ reasoning and orienting their ideas to one another. Additionally, professional 

development facilitators could work with teachers to examine together how to apply the macro 

and micro level strategies other teachers have found success with, in ways that are responsive to 

the classroom culture and equity goals of the teachers in the professional development. Findings 

from the case studies and PAR studies could also inform professional development for teacher 

educators, including mathematics coaches and teacher preparation methods course instructors. 

This professional development could include a similar design, sharing strategies for facilitating 

justification opportunities and thinking alongside the teacher educators about how to make sense 

of those strategies within their institutional contexts and goals.  

Professional development geared toward both audiences (teachers and teacher educators) 

could include practice-based methods for learning about justification opportunities and to 

practice facilitating them (Grossman et al., 2009). Ms. Josie found that watching examples of 

justification opportunities playing out in practice in other teachers’ classrooms, as well as 

practicing them herself (in a professional development and overtime in the classroom), was 

supportive in her own learning of facilitating justification opportunities. As referenced in 

Chapter 2, Josie mentioned,  

I think it's like anything, it's like part of it is seeing it in practice, and part of it is practicing. And 

the more you do it, the more of those small moments you'll recognize and know how to inject a 

justification opportunity into. So you have to do it a lot and you have to see somebody effectively 

do it to recognize it. You know, it's exposure, I guess.  

Other research has also found practice-based methods to be supportive for teachers in 

translating new or complex teaching practices that they hear about in theory (e.g., in courses or 

  99 

professional development) in a way that they can find both learnable to do themselves later and 

applicable in their own classroom contexts (Ball & Forzani, 2009; Kavanagh et al., 2020; 

Schiera, 2019 & 2021). Types of practice-based methods include representations, 

decompositions, and approximations of practice. Representations of the practice of justification 

could include video clips of teachers (e.g., from the case studies and PAR studies) facilitating 

justification opportunities. An important part of learning a new practice is seeing it play out in 

classrooms, particularly in similar and different ways by context, so that teachers can see what 

the practice looks like then debrief about how to make sense of the practice within their own 

classrooms. Decomposing justification opportunities could look like breaking down macro and 

micro level strategies for facilitating justification into more learnable parts, and discussing how 

the distinct parts make sense together as a whole. For example, the professional development 

could work with teachers to think of norms that could contribute to a classroom environment 

conducive to justification in their classrooms (a macro level strategy). Additionally, the 

professional development could decompose discussion facilitation strategies, such as pedagogy 

and discourse moves that decenters the teacher and positions students with authority of 

knowledge in ways that elicits and centers their ideas, discoveries, and reasonings. After seeing 

justification opportunities represented in practice and decomposing parts or types of justification 

opportunities, approximating the practice of justification can be supportive in helping teachers 

apply what they learned to practice facilitating justification opportunities themselves. Practicing 

in a mediated setting with the professional development facilitator could provide opportunities 

for teachers to try out strategies and get feedback, engage in do-overs, and debrief. Because 

justification opportunities are not prescriptive but look different across contexts and is a practice 

that is inherently responsive to student ideas, approximations of practice can support teachers in 

being more responsive later when trying out the practice with their actual students. Kavanagh 

and colleagues (2020) found that engaging in approximations of practice in mediated settings can 

support teachers in implementing the practice in ways that are more responsive to their students 

because of the time they had to practice beforehand. 

Closing 

In conclusion, this dissertation sought to understand how Josie provided opportunities for 

mathematical justification in a diverse, fourth-grade classroom to meet her equity goals for her 

students. Given that agency emerged as a prominent equity goal for Josie, analysis focused on 

 100 

the relationship between justification opportunities and her perceptions of the impact these 

opportunities had on students’ agency. To conclude my exit interview with Josie, I asked if there 

was anything else she would like the field to know. She enthusiastically offered her 

encouragement for more teachers to incorporate justification opportunities into their mathematics 

instruction, too, because of how engaged students become in the process. She shared, “Everyone 

should do it! Kids really like it! They're really good at explaining themselves.” Josie considers 

justification a “human skill,” as it can empower students’ agency within and beyond the 

mathematics classroom. She affirmed, “Because really, that's where we're leading them to is to 

be outside of our walls, and be free thinking humans out in the world.” 

 101 

 
 
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https://doi.org/10.1177/0022487109348479   

Bieda, K., Staples, M., Doherty, K., Matthews-Johnson, A., & Hwang, J. (2023). Considering the  

alignment between teachers’ conceptions of justification and their visions for equitable 
instruction. Research report in Proceedings of the forty-fifth annual conference of the 
North American Chapter of the International Group for the Psychology of Mathematics 
Education, vol. 1, 124-130. Reno, NV.  

Grossman, P., Compton, C., Igra, D., Ronfeldt, M., Shahan, E., & Williamson, P.  

(2009). Teaching practice: A cross-professional perspective. Teachers College Record, 
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Karunakaran, S. S., & Levin, M. (2022). Justification in the Context of Tertiary Mathematics:  

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J. (2020). Practicing responsiveness: Using approximations of teaching to develop 
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Varghese, M. (2018). Making justice peripheral by constructing practice as ‘‘core’’: How 
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