CONNECTING REFLECTION AND PRACTICE: TRANSFORMING A MATHEMATICS 

CLASSROOM CULTURE OF PARTICIPATION 

 
By 
 

Jose Manuel Martinez Hinestroza 

 

 

 

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 

 

 

 

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 

2019 

CONNECTING REFLECTION AND PRACTICE: TRANSFORMING A MATHEMATICS 

CLASSROOM CULTURE OF PARTICIPATION  

ABSTRACT 

 

 

 
By 
 

Jose Manuel Martinez Hinestroza 

Unlike previous research on participation in mathematics classrooms that focuses on the 

teacher’s role eliciting student talk, the purpose of this dissertation is to study the interactive 

process through which a teacher connected her reflection and practice, fostering transformations 

in the culture of participation in a third-grade Spanish immersion mathematics classroom. I draw 

on hermeneutic listening (Davis, 1996) to understand the teacher’s departure from listening for 

predetermined ways of participating. Methods from participatory design research with children 

(Bang & Vossoughi, 2016; Groundwater-Smith et al., 2014) and analytical tools from social 

semiotics (van Leeuwen, 2005) informed the collaborative design and analysis of interventions 

intended to promote a more inclusive culture of participation. Findings indicate that bringing the 

students’ and the teachers’ perspectives on participation into dialogue helped the teacher broaden 

her ideas on participation. A prolonged, iterative process was necessary to connect these 

reflections to a practice where the teacher learned to listen hermeneutically for participation. This 

process started with teacher-researcher co-designed interventions and continued as the teacher 

recognized participation in unexpected places and moments. The culture of participation evolved 

toward embracing multiple ways of participating in cooperative mathematical activity in which 

the students and the teacher addressed unanticipated ideas that emerged in interactions. I describe 

the semiotic innovations that characterized these transformations and the contextual influences 

that mediated semiotic innovations. I discuss implications for mathematics education research 

and professional development. 

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 
 
Copyright by 
JOSE MANUEL MARTINEZ HINESTROZA 
2019

 

ACKNOWLEDGEMENTS 

 

During this study, I learned about the importance of experiencing a sense of belonging 

and mutual influence in human activity, including mathematical activity. Luckily for me, I 

experienced belonging and mutual influence while developing this dissertation thanks to the 

unique ways in which my family, friends, and colleagues participated with me. Primero que 

todo, agradezco a mis papás, Luz Enith y Edilson, mis hermanos, Aura Enith y Edilson, y mis 

sobrinos Erika y Frederick, quienes participaron escuchándome, animándome, y ayudándome. 

To Roger, for celebrating my enthusiasm and joy when I was working with the children and the 

teacher, and for cheering me on with unwavering trust in my process. To my friends, Catalina, 

Elizabeth, Viviana, and Iván, for keeping me company and recognizing my effort. To Gloria and 

Juan Guillermo, for nurturing my passion for education and encouraging my curiosities. 

I thank many academic mentors who participated in this dissertation, too. To my advisor, 

Higinio Dominguez, who was a full participant in this work listening patiently, reading 

hermeneutically, and generously sharing the invaluable gifts that were his time and unique ways 

of thinking. To the other members of my dissertation committee, Tonya Bartell, Sandra Crespo, 

and Beth Herbel-Eisenmann, whose participation extended over time from the beginning of my 

graduate school experience when they kindly agreed to guide me. To India Plough, a mentor and 

a friend, who provided spaces in the Cultures and Languages across the Curriculum program in 

the Residential College in the Arts and Humanities where I could recharge, recover, and gain 

inspiration. 

Many thanks go to my colleagues and friends from the College of Education. To 

Abraham, who played an essential role, introducing me to a teacher he knew would appreciate 

 

iv 

my work and my presence in her classroom. Together with Abraham, my friends Christina, 

Romina, Yue, Bella and Gerardo, influenced my research by sharing the inner workings of their 

own powerful, provocative, and creative studies. To my writing group, Brittany, Andy, and 

Younggon, for providing a supporting and trustworthy space that motivated me to keep writing. 

And, of course, to the 21 students and the teacher who are the real heroes of this story and 

the ultimate participants in this dissertation. They made me feel that I belonged in their 

classroom, trusted me to influence their mathematics classes, and influenced my thinking and my 

teaching with their insightfulness and their warmth. Despite their surprise and disbelief when I 

said many educators needed to learn from what they were doing, they were remarkable teachers. 

Finalmente, la participación de mi hermano reta el tiempo porque influyó en mí desde 

que siendo niños me guiaba por el colegio, y reta el espacio porque su influencia inició en el 

cuarto que compartíamos y continúa hoy en el cielo desde donde me ayudó a escribir. 

 

 

 

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TABLE OF CONTENTS 

LIST OF TABLES ..................................................................................................................... viii 

LIST OF FIGURES ..................................................................................................................... ix 
Chapter 1 Introduction ................................................................................................................. 1 
Overview of Chapters .................................................................................................................. 4 
Chapter 2 Literature Review and Theoretical Framework ...................................................... 6 
Literature Review ........................................................................................................................ 6 
Previous views on participation in mathematics education research ....................................... 6 
The teacher's equalizing role .................................................................................................... 9 
The classroom culture of participation ................................................................................... 11 
Theoretical Framework .............................................................................................................. 14 
Participation as a complex situated phenomenon .................................................................. 14 
Hermeneutic listening for student participation ..................................................................... 16 
Chapter 3 Context of the Study and Methodology .................................................................. 19 
Site and Participants .................................................................................................................. 19 
The school system .................................................................................................................. 19 
The language immersion program .......................................................................................... 20 
The elementary school ........................................................................................................... 21 
The students ............................................................................................................................ 23 
The teacher ............................................................................................................................. 23 
Researcher role and positionality ........................................................................................... 24 
Preparing the Research Collaboration ....................................................................................... 25 
Methodology .............................................................................................................................. 26 
Participatory design research ................................................................................................. 27 
Research with children ........................................................................................................... 29 
Data Sources .............................................................................................................................. 30 
Reflections on what it means to participate ........................................................................... 30 
Connections between reflections on participation and practice ............................................. 32 
Social Semiotics Analysis of Connections Between Reflection and Practice ........................... 34 
Different meanings of participation ....................................................................................... 35 
Understanding transformations in the culture of participation ............................................... 36 
Collaborative Data Analysis ...................................................................................................... 37 
Analysis of perspectives on participation ............................................................................... 37 
Analysis of connections between reflection and practice. ..................................................... 39 

Chapter 4 Broadening the Teacher’s and Researcher’s Perspectives on Participation ....... 41 
Learning to See Participation as Qualifiable ............................................................................. 42 
Relocating Participation as a Dynamic Interaction Between the Individual and the Social ..... 50 
Rethinking the Teacher’s Role in Participation ......................................................................... 57 
Reflections on My Evolving Perspectives on Participation ...................................................... 64 

 

vi 

Chapter 5 Preparing to Recognize Diverse Participation ....................................................... 66 
First Intervention Cycle: Addressing Aspects that Stifled Participation ................................... 67 
Focal strategy I: Addressing wonderment questions .............................................................. 69 
Focal strategy II: Gallery walks ............................................................................................. 70 
Second Intervention Cycle: Recognizing Multiple Ways of Participating ................................ 70 
Focal strategy: Spotlighting gestures, drawings, and movement ........................................... 73 
Third Intervention Cycle: Encouraging Collaboration Among Groups .................................... 73 
Focal strategy: Group ambassadors ........................................................................................ 75 

Chapter 6 Connecting Reflection and Practice ........................................................................ 77 
Expanding the Scope of Mathematical Ideas Available for Exploration .................................. 78 
Addressing students’ curiosities ............................................................................................. 79 
Recognizing mathematical ideas in “off-task” behavior ........................................................ 84 
Using Multiple Semiotic Resources to Develop Mathematical Ideas ....................................... 89 
Fluid Participation in Mathematical Activity ............................................................................ 95 

Chapter 7 Discussion: What It Took to Transform the Culture of Participation .............. 104 
Reconsidering What Counts as Participation .......................................................................... 104 
Listening Hermeneutically for Participation ........................................................................... 106 
Expanding Research on Participation ...................................................................................... 108 
Reimagining the Teacher as a Learner of Student Participation ............................................. 110 
APPENDICES ........................................................................................................................... 113 
APPENDIX A—Initial Focus Group Questionnaire ............................................................... 114 
APPENDIX B—Initial Teacher Interview Questionnaire ....................................................... 115 
APPENDIX C—Fractions Task .............................................................................................. 116 
REFERENCES .......................................................................................................................... 117 
 
 
 

 

 

vii 

LIST OF TABLES 

 
 
Table 3.1 Number of classrooms per grade .................................................................................. 22 

Table 4.1 Example of participation as qualifiable and as quantifiable ......................................... 43 

Table 4.2 Examples of problematizing participation as talk ......................................................... 47 

Table 4.3 Example of over-competitiveness in whole class discussions ...................................... 51 

Table 4.4 Example of participation as belonging to a group ........................................................ 55 

Table 4.5 Example of participation in different activities ............................................................ 58 

Table 4.6 Example of pacing participation ................................................................................... 61 

Table 6.1 Valery's previous approach to topics to explore ........................................................... 78 

Table 6.2 Example of wonderment question ................................................................................ 80 

Table 6.3 Calum's sense of participation ...................................................................................... 81 

Table 6.4 Willie's strategy for the area of circles .......................................................................... 85 

Table 6.5 Students' initial strategies to compare fractions ............................................................ 90 

Table 6.6 Juan's strategy for the area of a circle ........................................................................... 97 

Table 6.7 Halley's strategy for the area of a circle ........................................................................ 99 

 

 

 

viii 

LIST OF FIGURES 

 
 
Figure 6.1 Emi's gestures on the graph ......................................................................................... 92 

Figure 6.2 Juan’s circles ............................................................................................................... 96 

Figure 6.3 Figuring out the area of a circle ................................................................................. 100 

 

 

 

ix 

Chapter 1 Introduction 

The topic of my dissertation was motivated by a three-year collaboration with Valery 

Abad (all names are pseudonyms) in her third-grade mathematics class. Valery and I discussed 

and tried different approaches to respond to her interest in what she called “abrir la 

conversación” [to open up discussions]. The issue of not all students participating became a 

recurrent concern in our discussions. For example, after one lesson where the teacher and I 

thought most students had been noticeably enthusiastic, addressing each other during a whole-

class discussion, Valery mentioned that Arnie’s participation had puzzled her because of how 

quiet he had been. As I was leaving the school that day, I saw Arnie outside waiting for his 

parents to pick him up. Without any prompting, Arnie told me how much he had enjoyed the 

class and that he thought one of his classmate’s strategies during the number talk was very smart 

(field note, December 6, 2016). 

When I shared Arnie’s comment with Valery, our conversation shifted from whether 

Arnie participated to how he participated. Specifically, Valery mentioned he tended not to speak 

up in whole-class discussions but he seemed to be “un buen oyente” [a good listener]. Valery 

said that the idea of speaking up in class could be off-putting for Arnie (field note, December 8, 

2016). Observations and discussions like these motivated me to question what it means to 

participate in a mathematics classroom. Specifically, I began to wonder how students’ and 

teachers’ ideas about participation may differ, and how teachers’ ideas of what it means to 

participate may shape what counts as participation. 

Research on participation in mathematics classrooms has primarily studied the role of the 

teacher in eliciting student talk. Few studies have focused on how students actually participate 

and what they have to say about their own participation (Dallimore, Hertenstein, & Platt, 2004; 

 

1 

Kane & Maw, 2005). The tendency to overlook students’ perspectives in favor of teachers’ and 

researchers’ perspectives is particularly marked in studies involving elementary school children 

(Groundwater-Smith, Dockett, & Bottrell, 2014). My dissertation addresses this issue by drawing 

on students’ perspectives on participation to support a teacher reflect on what counts as 

participation and adjust her teaching accordingly. Children’s perspectives challenged adults’ 

initially narrow conceptualizations of participation. 

The purpose of this dissertation is to study the interactive process through which a 

teacher connected her reflection and practice, fostering transformations in the culture of 

participation in a third-grade Spanish immersion mathematics classroom. I focus on the 

following research question: 

How did a teacher connect her reflection on what it means to participate in mathematical 

activity with her practice in order to transform the classroom culture of participation? 

I focus on this interactive connection between reflection and practice to make sense of how a 

classroom culture of participation evolved. 

Sociocultural learning theory (Rogoff, Paradise, Mejía Arauz, Correa-Chávez, & 

Angelillo, 2003; Vygotsky, 1978; Wenger, 1998) informs my conceptualization of participation. 

I focus on the classroom culture of participation as a situated phenomenon where the cultural 

system that the classroom constitutes influences what is considered legitimate participation, 

including which ways of participating are overlooked or dismissed. Rather than promoting a 

culture of participation that imposes the teacher’s perspectives or the students’ perspectives on 

participation, I focus on transformations in the culture of participation grounded on mutually 

informed reinterpretations of these perspectives. These reinterpretations involve the teacher’s 

commitment to reflecting on and questioning preconceived ideas about participation. To 

 

2 

understand these reinterpretations, I bring hermeneutic listening (Davis, 1996; 1997) to the study 

of participation. Hermeneutic listening attends to the transformation from rigid and preset 

mathematics teaching to embracing unanticipated and surprising ways of engaging with 

mathematical ideas (Davis, 1996). In addition to guiding my exploration of how the class 

reinterpreted what counts as participation, this framework also informed the interventions 

designed to foster a classroom culture of participation inclusive of diverse ways of participating. 

A Participatory Design Research approach (Bang & Vossoughi, 2016) and methods from 

research with children (Veale, 2015; Groundwater-Smith et al., 2014; Punch, 2002) informed the 

design and analysis of these interventions. Analytical tools from social semiotics (Halliday, 

1978; O'Halloran, 2005; van Leeuwen, 2005) informed the analysis of the transformation of the 

culture of participation of this classroom. 

This dissertation contributes new knowledge about mathematics research and practice. 

My study involves both the students’ and the teacher’s perspectives in the conceptualization of 

participation and in promoting and making sense of the transformation of the classroom culture 

of participation. In doing so, my study helps us reconsider what it means to participate and 

whose perspectives on participation should inform our research. This involves practice speaking 

to theory, as definitions of participation emerged from and considered participants’ specific 

context, needs, and experiences. My study also expands previous research on teachers’ roles on 

participation by informing how teachers can learn from students to transform the culture of 

participation. This involves connecting teacher reflection and practice, highlighting a teacher’s 

commitment and support to question preconceived notions on participation, and creative teaching 

strategies to embrace and support complex and diverse practices that students bring to the 

 

3 

classroom. Ultimately, this dissertation contributes an approach to mathematics teaching and 

research that honors student diversity. 

Overview of Chapters 

In the next chapter, I situate this study in the literature on student participation in 

mathematics classrooms, contrasting prevailing views on participation as student talk with the 

conceptualization of participation as a complex situated phenomenon that guides this 

dissertation. I discuss hermeneutic listening as a framework informing the teacher’s 

reinterpretation of situated participation. In Chapter 3, I describe the research design, including 

how the participants and I prepared our research collaboration. Drawing from Participatory 

Design Research and principles from research with children, the teacher and I designed 

intervention cycles as part of a methodology that supported us to elicit students’ perspectives on 

participation. I introduce social semiotics as an analytical framework consistent with both the 

theoretical framework and the purpose of the study. Closing this chapter is a description of the 

data generation and analysis. Chapters 4, 5 and 6 present findings at the level of reflection, 

practice, and reflection-practice connections, respectively. Chapter 4 presents findings about the 

teacher’s initial reflections on participation, as she brought into dialogue the students’ and her 

perspectives on what it means to participate. Chapter 5 describes findings regarding the initial 

process of preparing to recognize multiple ways of participating supported by the iterative co-

design of flexible interventions. Chapter 6 focuses on findings about the ongoing connections 

between reflection and practice that helped the teacher engage in hermeneutic listening for 

participation to transform the culture of participation. Finally, in Chapter 7 I bring into my final 

discussion the three levels of findings with a focus on the connections between reflection and 

 

4 

practice that promoted the transformations of the culture of participation. I address implications 

for research and practice. 

 

5 

Chapter 2 Literature Review and Theoretical Framework 

This study is situated on the literature on participation in mathematics classrooms, with a 

focus on how participation has been conceptualized. I pay particular attention to 

conceptualizations of the role of the teacher and the classroom culture on participation. I then 

define participation from a sociocultural theory of learning and discuss how I draw on 

hermeneutic listening to explore the teacher’s shift toward transforming her understanding of 

participation. 

Literature Review 

Previous research on participation in mathematics classrooms has mainly equated 

participation with student talk. Accordingly, the role of the teacher has frequently been 

characterized in terms of attempting to distribute talk time uniformly among students. Here, I 

review these previous views on participation and then discuss the classroom culture of 

participation as a concept that focuses on social aspects of participation, including meanings 

associated with participation that are simultaneously shared and contested. 

Previous views on participation in mathematics education research. Participation is 

nearly equated with student talk in mathematics education research. Many studies use the terms 

participation and student talk interchangeably (For a few examples see Brown, 2017; Franke et 

al., 2015; Ing et al., 2015; Jansen, 2006; Krummheuer, 2007; O'Connor, Michaels, Chapin, & 

Harbaugh, 2017; H. Smith & Higgins, 2006; Takeuchi, 2015; Webb et al., 2017; 2014). For 

example, when studying students’ participation in mathematics classrooms Jansen (2006) 

focused on students “talking about mathematics” (p. 412). Similarly, when examining how social 

structures influence student participation in mathematics classrooms, Civil and Planas (2004) 

focused on how some student talk influences how classes unfold “while other voices are not 

 

6 

heard” (p. 8). Most studies that acknowledge diverse ways of participating in mathematics 

classrooms do so on a parenthetical note preceding the announcement of the studies’ focus on 

student talk (Foote & Lambert, 2011; Reinholz & Shah, 2018; M. B. Wood & Hackett, 2018). 

The few studies that have explored participation beyond student talk consider listening (Lack, 

Swars, & Meyers, 2014; Moschkovich, 2018), reflecting (Rodd, 2003), gesturing (Williams-

Pierce et al., 2017), and drawing (O'Halloran, 2015). Although student talk is an important 

dimension of participation in mathematics classrooms, the overemphasis on student talk ignores 

less vocal but equally relevant ways of participating. In this study, I consider these and other 

spontaneous ways of participating, acknowledging students’ multiple ways of being part of 

mathematical activity. 

Previous research that views participation as student talk frequently focuses on this type 

of participation occurring in whole class discussions (Baxter & Woodward, 2001; Esmonde & 

Langer-Osuna, 2013; Lo, Wheatley, & Smith, 1994; Lubienski, 2000). These studies highlight 

differences in the amount of student talk during discussions, exploring individual and classroom 

characteristics that prevent certain students from speaking up. In this dissertation, in addition to 

exploring multiple ways in which participation occurs, I also explore multiple types of activities 

where it takes places. Specifically, I explore participation in whole class discussions, as well as 

in individual and small group work. 

Mathematics education research has also frequently treated participation as a quantifiable 

individual phenomenon. In some cases, this quantification is framed as absence or presence of 

participation. For example, when studying differences between male and female students’ 

participation, Fassinger (1995) stated that “some students participate daily and others by the end 

of the semester have not uttered a word” (p. 82). Similarly, Foote and Lambert (2011) 

 

7 

problematized some students’ “lack of participation” (p. 248). The quantification of participation 

is concerned with increasing the number of utterances per student (Foote & Lambert, 2011; H. 

Smith & Higgins, 2006) or “the amount of detail students provide” (Webb et al., 2017, p. 6). 

One assumption of this view on participation that is rarely interrogated is that more 

student talk is better. The locus of participation is assumed to be in individual students. Students 

who talk more are active and gain opportunities to learn, and those who talk less are judged as 

passive and in need of more opportunities to speak up (Baxter & Woodward, 2001; Lack et al., 

2014; Webb et al., 2014). Given the limited time available for students to talk, one student’s 

participation means other students’ non-participation (Hardy, 2007; Moschkovich, 2018; Skott, 

2013; Wager, 2014; M. B. Wood & Hackett, 2018). In this dissertation, I contribute to alternative 

conceptualizations that, rather than determining who participates more and who participates less, 

consider different ways in which student participation may occur simultaneously as one student’s 

participation interacts with others’. Although some students may benefit from expanding their 

ways of participating, there is the risk of forcing quieter students to come closer to what may be 

perceived as a norm for amount of talk based on their outspoken peers. Instead, I focus on 

understanding the complex and diverse ways in which participation unfolds during mathematical 

activity. This consideration challenges normative views of participation as a predetermined kind 

and quantity of student talk and opens up possibilities for the class to recognize ideas that may 

otherwise go unnoticed. 

In addition to exploring who participates and how, previous research has also explored 

what students participate in. The focus has been on specific mathematical practices established 

by the Common Core Standards for Mathematical Practice (Common Core State Standards 

Initiative, 2010) and NCTM’s mathematical processes (National Council of Teachers of 

 

8 

Mathematics NCTM, 2000). These practices include articulating problem-solving strategies, 

making and justifying claims, attending to precision, and attending to others’ ideas (Brown, 

2017; Esmonde & Langer-Osuna, 2013; Franke et al., 2015; Krummheuer, 2007; Moschkovich, 

2018; Turner, Dominguez, Maldonado, & Empson, 2013). In addition to these mathematical 

practices, in this dissertation I consider practices that may emerge in the context of diverse ways 

of participating, such as posing problems (Dominguez, 2016), listening (Goldin, O'Neill, Naik, & 

Gomez Zaccarelli, 2019), and helping others (Jansen, 2006). I also consider the specific content 

of the mathematical tasks the class participates in. Some of this content emerges unexpectedly 

and may deviate from the predetermined curriculum. 

The teacher's equalizing role. Previous research on mathematics education has 

characterized the teacher’s role regarding participation mainly in terms of eliciting student talk. 

Specifically, teachers strive to uniformly distribute talk time among students, disrupting the 

tendency to attend to mathematical ideas uttered by a few dominant students. This role involves 

developing classroom norms conducive to equitable student talk (Franke et al., 2015; Lack et al., 

2014; Wager, 2014; Webb et al., 2014). As Webb et al. (2014) put it, “Techers can set ground 

rules and guidelines for desired participation, promoting greater student talk“ (p. 81). These 

norms influence student talk by signaling to students the desirable ways to add to discussions. 

To foster these classroom norms, teachers intervene during discussions so that the class 

hears from students who usually do not speak up. Some associated teacher moves include 

inviting quiet students’ contributions (O'Connor et al., 2017; Wager, 2014); asking probing 

questions (Franke et al., 2015; Ing et al., 2015); validating marginalized students’ mathematical 

ideas (Turner et al., 2013; M. B. Wood et al., 2019); legitimizing the use of multiple languages 

(Takeuchi, 2015); using multiple interaction structures, such as whole-class discussions, small-

 

9 

group work, and student-student conversations (Webb et al., 2017); and guiding students’ 

reflection on their own participation (Lack et al., 2014). 

An argument that these studies develop is that the teacher practices that elicit 

participation cannot be prescribed (Esmonde & Langer-Osuna, 2013; Franke et al., 2015; Skott, 

2013). Rather than preplanning a sequence of specific moves, teachers elicit participation when 

they responsively draw on this repertoire of moves as they mediate classroom interactions. 

Teachers observe for patterns in students’ amount of talk and whose utterances the class is 

considering. Teachers draw on these observations and their knowledge of their class to decide on 

moves that encourage specific students to voice their ideas. Over time, students pick up on those 

moves and become aware of who is expected to talk and when (Hardy, 2007). When teachers fail 

to develop these classroom norms, a cycle is perpetuated where a few outspoken students 

monopolize discussions while the ideas of quieter students remain unheard. 

In addition to drawing on specific teacher moves, teachers foster participation-related 

classroom norms through the tasks that they implement. Previous research has argued that 

teachers can elicit talk by piquing student interest through mathematically challenging tasks 

(Emanuelsson & Sahlström, 2008; Jansen, 2006; Lo et al., 1994). When these tasks require 

interaction, they promote the need for different students to contribute to discussions (Boaler, 

1999; Cohen, 1994; Emanuelsson & Sahlström, 2008; Featherstone et al., 2011). In this way, the 

teacher orchestrates students’ opportunities to contribute to mathematical discussions. Even 

when these types of tasks are in place, however, the teacher still observes during small-group 

work, drawing on a repertoire of teaching moves when necessary to equalize the amount of 

student talk. 

 

10 

Instead of focusing on the role of the teacher as equalizing the amount of student talk, in 

this dissertation I focus on connections between reflection and practice that are foundational to 

fostering an inclusive culture of participation. Connecting reflection that destabilizes taken for 

granted perspectives on participation with a practice that honors student participation involves 

teachers assuming the role of learners of what it means to participate in a specific classroom. In a 

bidirectional, iterative process, practice informs further reflection to make sense of the evolving 

culture of participation. I explore a teacher’s process of connecting reflection and practice that 

helped her see participation in spaces and moments that were hiding from her perspective. 

The classroom culture of participation. Educational researchers have theorized culture 

as fluid, dynamic and pluralistic (Ladson-Billings, 2014; Paris, 2012; Paris & Alim, 2014). This 

theorization challenges the tendency to attribute predetermined static cultural traits to specific 

groups (for example, Latinos or women). Rather than assuming a monolithic and static culture 

associated with specific races, ethnicities or languages, this theorization acknowledges that 

students are members of several communities with distinct cultures and evolving practices. 

Students bring these multiple cultural ways of being and practices to the classroom. The role of 

the teacher involves understanding, honoring and exploring students’ multiple cultural practices, 

instead of attempting to replace them with practices from the dominant culture of the school or 

the discipline (Paris, 2012). This theorization of culture fosters pluralism and orients teachers 

“toward the idea that learners can be sources and resources of knowledge and skills” (Ladson-

Billings, 2014, p. 79). In mathematics education, Louie (2017) drew on this theorization to argue 

that teachers unintentionally exclude students who do not align with the dominant culture of 

mathematics education from learning opportunities. She called for more research on how 

teachers may disrupt this culture of exclusion. This theorization informs my conceptualization of 

 

11 

a mathematics classroom culture. The culture of a classroom, like culture at other social levels, is 

heterogeneous and participants value different practices and ways of making sense of 

mathematical ideas (Gresalfi, Martin, Hand, & Greeno, 2009; Moschkovich, 2018; T. Wood, 

Williams, & McNeal, 2006). Teachers and students continuously reconfigure these values during 

mathematical activity (Bishop, 1988; 1991; Cobb & Yackel, 1998; Nasir, Hand, & Taylor, 

2008). This conceptualization brings attention to the interaction between the individual and the 

social: individuals influence the development of the class culture and the class culture influences 

how individuals engage in activity (Gresalfi et al., 2009; Hodge & Cobb, 2016; Lo et al., 1994; 

Nasir et al., 2008). 

I refer to the culture of participation of a mathematics classroom as the relatively shared, 

yet continuously negotiated, sense of what counts as participation. Previous research suggests 

that teachers promote a classroom culture that elicits and supports a specific type of participation 

through explicit classroom norms and through favoring ideas that emerge in particular ways 

(Civil & Planas, 2004; Nickson, 1994; O'Connor et al., 2017). Over time, students learn to 

recognize the ways of participating that the teacher values (Hardy, 2007; Skott, 2013; Turner et 

al., 2013). While some students align their participation to match teacher’s expectations, others 

do not (Hardy, 2007; Hodge & Cobb, 2016; Lubienski, 2000). 

As discussed above, teachers frequently attend to the ideas of the most vocal students, 

missing out on others’ ideas. In this process, a class culture of participation that overvalues 

student talk gets reinforced. In this dissertation, instead of attempting to change how individual 

students participate to match the teacher’s expectations, I explore a way of broadening what 

counts as participation by focusing on understanding the multiple meanings that participation has 

for students and teachers. In turn, considering mathematical ideas that emerge through these 

 

12 

multiple ways of participating may contribute to challenging the overvaluing of student talk as 

the ultimate form of participation. 

Researchers have called for the exploration of insiders’ perspectives on participation to 

inform the development of an inclusive classroom culture (Brown, 2017; Hodge & Cobb, 2016; 

Pimm, 1994). Previous studies have focused primarily on teacher’s insights (as argued by 

Esmonde & Langer-Osuna, 2013; Lack et al., 2014). Few studies have asked students about their 

own participation in mathematics classrooms (for exceptions, see Jansen, 2006; Lack et al., 

2014; Turner et al., 2013). These studies suggest that students who see their participation as 

influencing the ideas the class considers see themselves as competent. Conversely, those whose 

ways of participating are not valued or not acknowledged see themselves as less competent 

(Lack et al., 2014). These studies also suggest that just as teachers influence the culture of 

participation, so do students, if we care to learn from what they have to say. For example, Jansen 

(2006) discusses the potential implications on teaching of considering students’ willingness to 

participate in specific ways and the types of activity they prefer to participate in. My contribution 

to this line of research consists of gaining a deeper understanding of students’ perspectives on 

participation and drawing on their perspectives to inform the transformation of the classroom 

culture of participation. 

The study of the students’ and the teacher’s participation-related perspectives and 

practices implies a focus on the context-specific nature of participation. What counts as 

participation may differ across classrooms (Hodge & Cobb, 2016; Pimm, 1994). Moreover, the 

school and social context where the classroom is situated may influence the participation notions 

that are valued, as well as teachers’ and students’ perspectives on participation (Civil & Planas, 

2004; Esmonde & Langer-Osuna, 2013; Lubienski, 2000; Matthews, 1984). This situated nature 

 

13 

of participation is consistent with the previous discussion on the impossibility of predetermining 

teacher moves to promote participation. Accordingly, in this dissertation I explore how the 

teacher’s reconceptualization of participation continued to develop in dialogue with the evolving 

culture of participation. 

Theoretical Framework 

In mathematics education research, a predominant conceptualization of participation 

highlights student talk. In this dissertation, I expand this conceptualization by considering time 

and space as two relevant dimensions of participation. This means locating participation within 

longer stretches of time, in more varied spaces, and across a myriad of artifacts so as not to miss 

what participation means for students. I begin by discussing the theoretical grounding of my 

conceptualization of participation and how I use this concept to understand the culture of 

participation in a mathematics classroom. 

Participation as a complex situated phenomenon. I draw on sociocultural theory to 

define participation as an evolving social experience of engaging in meaningful shared activity, 

that combines doing, talking, thinking, feeling, and belonging (Lave & Wenger, 1991; Rogoff et 

al., 2003; Vygotsky, 1978; Wenger, 1998). Participation is a situated phenomenon in which the 

individual and the social interact as members of a bounded social community, such as a 

classroom, coordinate the pursuit of a shared enterprise. It is situated because the context 

influences both the activities that are considered meaningful and the ways of becoming part of an 

activity (R. Thomas, Whybrow, & Scharber, 2012b). The shared history and culture of the 

community also influence how different ways of participating are valued (Mafra Goulart & Roth, 

2006; Rogoff et al., 2003; Wenger, 1998). Participation involves an interaction between the 

individual and the social because, rather than being a static set of practices, members of a 

 

14 

community permanently negotiate what it means to participate (Wenger, 1998). As the 

community develops over time, the activities in which members engage and how they participate 

in these activities evolve, too. Different members may introduce novel activities or ways of 

engaging in activity. When an individual’s ways of participating are legitimized and influence 

how others participate, the member experiences a sense of belonging. 

An implication of this conceptualization is that participation involves developing a sense 

of belonging. Engaging in shared activity constitutes participation when students feel they are 

influencing the activity (R. Thomas, Whybrow, & Scharber, 2012a; 2012c; Wenger, 1998). For 

example, although some teachers may dismiss silence as a form of participation, silent students 

may be joining in shared activity through intent observation and thought (Moschkovich, 2018; 

Rogoff et al., 2003; Schultz, 2010). When studying participation in terms of mutuality and 

influence, von Wright (2006) argued that students feel they are participating when they perceive 

others as taking their ideas seriously. Similarly, students may not feel they are participating even 

if they engage in talk, unless they feel others consider their contributions. In this 

conceptualization of participation, what matters is not so much a student’s individual 

contribution at a specific point in time but the impact that such contribution may have over time 

and across the complex space constituted by the classroom . Students’ recognition of their own 

sense of participation highlights my interest in eliciting students’ perspectives on participation. 

This conceptualization also has implications on the role of the teacher. What counts as 

participation varies from classroom to classroom (Anderson, 1998; Hammel et al., 2009; Hickey, 

2003). Even within each classroom, teachers’ and students’ meanings of what counts as 

participation may differ (Mafra Goulart & Roth, 2006; Moguel, 2004; Rocca, 2010). Teachers 

may decide to focus on helping students participate a certain way. Although this approach has 

 

15 

the potential of helping students expand the ways in which they participate, it can also exclude 

children whose preferred ways of participating are overlooked. In contrast, when the classroom 

allows multiple ways of participating to influence activity, the community expands the ways in 

which students may make sense of ideas and more students experience a sense of belonging in 

mathematical activity (Dallimore et al., 2004; Davitti & Pasquandrea, 2017; Norton, 2001). In 

this dissertation, the teacher’s role includes connecting this reconceptualization to her practice by 

fostering a transformation of the culture of participation grounded on students’ actual ways of 

participating. This conceptualization also pays attention to ways of participating that are spatial 

an temporal in nature, including gestures, gaze, movement, writings, and drawings. I consider the 

role of the teacher as someone learning to see participation in multiple locations and across time. 

Hermeneutic listening for student participation. As discussed before, rather than 

focusing on a teacher-imposed view of what participation is, my study focuses on a teacher-

researcher collaboration to help the teacher reconceptualize what counts as participation. To 

guide the study of this reconceptualization, I draw on hermeneutic listening’s attention to 

“imaginative participation in the formation and transformation of experience through an ongoing 

interrogation of the taken-for-granted” (Davis, 1996, p. 53). Hermeneutic listening refers to a 

concerted and participatory type of interaction through which teachers and learners influence the 

unfolding of classroom possibilities (Davis, 1996; 1997). Hermeneutic listening is a departure 

from listening for predetermined mathematical understandings expressed in preconceived ways. I 

bring this concept to understand a teacher’s departure from listening for predetermined ways of 

participating, attending to the multiple situated ways in which students participate. 

Adding to previous research on mathematics education that has drawn on hermeneutic 

listening (Arcavi & Isoda, 2007; Crespo, 2000; Doerr, 2006; Hintz & Tyson, 2015; Johnson & 

 

16 

Larsen, 2012), I focus on three concepts that are relevant to the exploration of participation as a 

situated phenomenon: listening to more than spoken language; the teacher as a full participant in 

the learning activity; and listening as involving both the individual and the social. First, 

hermeneutic listening transcends spoken language as it involves attending “not just to the voices 

and actions of students but also to the… cultural situation in which the teaching is located” 

(Davis, 1997, p. 374). I draw on hermeneutic listening to interpret students’ embodied actions, 

silence and reflection as possible ways of participating in mathematical activity. 

Second, hermeneutic listening positions the teacher as a full participant in the learning 

activity. Instead of conceptualizing the teacher as a transmitter of knowledge or a removed 

facilitator of student activity, both teacher and students question, interpret and transform the 

direction of a lesson (Davis, 1997). The teacher does not engage in surveillance of students to 

ensure they enact specific actions—such as formal talk—and not others (Davis, 1996). The 

teacher joins students in interpreting mathematical ideas and redirecting collective action 

accordingly. Situating the teacher as an active participant requires that the teacher no longer 

searches for predetermined ways of speaking about mathematical concepts, but instead interprets 

and responds to students’ ideas developed and expressed in spontaneous and diverse ways. 

Third, hermeneutic listening is concerned with the individual and the social. In line with 

my focus on the transformation of the culture of participation of the classroom, hermeneutic 

listening involves attending to how the social group influences what individuals do and vice 

versa (Davis, 1997). The interdependence of agents and setting, self and other, and individual 

and collective draws attention to how the collective moves across ideas and actions. That is, the 

focus is on the possibility of interpretations of classroom phenomena that facilitate the 

emergence of mathematical ideas. Additionally, hermeneutic listening involves a commitment on 

 

17 

the part of the researcher and the teacher to engaging with students in exploring mathematical 

ideas. 

Hermeneutic listening applied to the study of participation draws attention not only to 

formal ways of expressing ideas but also to divergent and creative ways of engaging in 

mathematical activity. Transforming the classroom culture of participation toward attending to 

alternative ways of participating involves students, the teacher and the researcher coming 

together to listen for difference (Davis, 1997). Interactions in which participants listen for 

difference are grounded on a teacher’s belief in students’ ability to generate mathematical ideas, 

and a commitment to approaching students’ ideas with authentic curiosity (Davis, 1997). 

Teaching strategies that operationalize hermeneutic listening include: asking genuine 

questions—questions for which the teacher does not know the answer—, directing students’ 

listening—drawing attention to what others do and say—, and amplifying students’ 

participation—building on what students do and say to inform how lessons unfold—(Hintz & 

Tyson, 2015). These strategies informed the collaborative design of interventions intended to 

support the transformation of the culture of participation. 

The above theoretical commitments conceive of participation as a complex situated 

phenomenon that can be understood through hermeneutic listening. Aiding the process of 

recognizing participation more broadly are the ongoing collaboration to raise perspectives on 

participation to awareness, the evolving role of the teacher, the types of lessons and 

mathematical activities, and the flexible adjustment of tasks, lessons and units in response to 

student participation. Considering these contributing factors sheds light on how the teacher 

learned to recognized unexpected situated participation and how she fostered an inclusive culture 

of participation. 

 

18 

Chapter 3 Context of the Study and Methodology 

In this chapter, I discuss the participatory-design research with children methodology and 

the social semiotics analytical framework that guided the study of the transformation of the 

culture of participation in this mathematics classroom. I begin describing the research context. 

Site and Participants 

The school system. This K-12 school system consists of five campuses located in 

western Michigan. More than 2,200 students from various Christian denominations attend these 

religious schools, founded almost 100 years ago. A quarter of the student population is non-

White. The vision and mission statements mention a commitment to expanding the diversity of 

the student body, with a focus on increasing affordability for students from diverse 

socioeconomic backgrounds. The schools’ financial aid program grants partial scholarships to 

50% of the student population. Additionally, through a variable tuition system, what each family 

pays depends on their unique financial situation. The school system attributes its high (94%) 

retention rate in part to these efforts toward affordability. More than 90% of high school 

graduates enter college, and many of them received a partial scholarship from the school system. 

Teachers and staff periodically attend professional development sessions on how to serve 

students from diverse cultural, ethnic, denominational and socioeconomic backgrounds. Teachers 

also engage in professional development on the principles and practices of the Learning without 

Limits project developed by scholars in the Faculty of Education at the University of Cambridge 

(Hart, Dixon, Drummond, & McIntyre, 2004). This professional development focuses on shifting 

away from ideas of student ability as a fixed individual characteristic to the notion of 

transformability. Transformability refers to a conviction that teachers and students can develop 

transformational learning opportunities. To help students take up these opportunities, teachers are 

 

19 

expected to communicate high expectations for all students and to help students experience a 

sense of confidence when putting effort into classroom activities. These teaching practices 

include asking questions that foster student reasoning, and publicly acknowledging students’ 

effort and perseverance, and rejecting ability-related labels and stratified grouping arrangements. 

Common Core State Standards, the Michigan Curriculum Framework, and the school 

system’s goals for graduates inform curriculum and assessment design. Goals for graduates are 

intended to permeate teaching at all levels, and they include developing creative problem 

solving, teamwork, and effective communication skills. In line with the schools’ efforts toward 

diversity, goals for graduates include demonstrating a willingness to engage with a range of 

people and ideas in respectful ways and to become justice seekers actively involved in 

community issues. Teachers are encouraged to weave these goals in their lessons and to 

explicitly discuss these goals with students. 

The language immersion program. Since 2010 and in response to the interest of several 

parents, the school system offers a Spanish language immersion program. The goals of the 

immersion program are to support students to simultaneously develop functional proficiency in 

Spanish, cultural competence, and mastery of grade level curricular standards. According to a 

survey the school administers to parents annually, the main motivations for enrollment in the 

immersion program are for students to eventually enjoy the professional opportunities of 

speaking a language other than English, and to encourage cross-cultural understanding. The 

majority of the children of Latino Spanish-speaking parents in the school system attend the 

immersion program. These children represent 20% of students in the immersion program and 

their parents choose the program mainly to support their children’s bilingualism and 

biculturalism. 

 

20 

The program follows a language enrichment full immersion model (Brisk, 2011) where 

all instruction is in Spanish in the beginning grades and some classes are taught in English in 

upper elementary grades. From PK to first grade all instruction is in Spanish. Second to sixth 

grade include 60 minutes a day of instruction in English. Mathematics is taught in Spanish, one 

hour a day, five days a week. PK-5 students have a class that focuses on Spanish grammar and 

communicative skills, and on the traditions and history of Spanish speaking countries. After fifth 

grade, students have the option of joining an immersion cohort that offers Spanish language arts, 

social studies, and Bible studies in Spanish. For all other classes, students join the mainstream 

English-speaking classrooms. 

Students in the immersion program frequently engage in activities with the local Spanish-

speaking community. This includes field trips to local organizations, attending cultural events, 

interacting with guest speakers, engaging in community outreach, and engaging with events that 

involve the Spanish-speaking community. All teachers and teacher aides in the language 

immersion program are native or near-native speakers of both Spanish and English. Teachers are 

encouraged to use Spanish at all times and to use their discretion to identify moments when 

students’ emotional or physical well-being may require the use of English. 

The elementary school. The elementary school serves nearly 650 PK-4 students, with an 

average class size of 22 students. There are 91 faculty and staff members. The Spanish 

immersion program for the entire school system originated in this school. Table 3.1 shows the 

number of classrooms per grade, including the number of Spanish immersion classrooms, for the 

2017-2018 academic year. In addition to homeroom teachers, there are teachers for the co-

curricular classes (arts, music, physical education, and Spanish). The school principal encourages 

collaboration among teachers and among teachers, coaches, and subject specialists. These 

 

21 

collaborations include co-planning lessons, peer observation, and co-developing co-curricular 

activities and events. The school also encourages collaboration with the teacher preparation 

program at a local college. During the spring semester, some teachers serve as mentors to student 

teachers, including Spanish-English bilingual student teachers placed in the immersion 

classrooms. 

 

The school provides pacing guides that include the sequence of topics for each teaching unit, 

the beginning and finishing dates for the unit, and standards and learning goals. Each teacher 

decides how to plan lessons, the activities to include and the amount of time devoted to different 

topics within the unit. The school selected the textbook that teachers are expected to use for 

mathematics lessons. The textbook includes a teacher’s version with lesson plan ideas and 

materials, and there are English and Spanish editions, so both immersion and mainstream 

classrooms use the same textbook. Each teacher independently designs on-going assessments for 

the content area. Teachers from each grade level get together to design an end-of-unit test. 

Immersion teachers translate the test to Spanish. Each teacher decides when to administer the test 

within the dates established for the unit. The school also administers an end-of-year test for 

reading and math, designed by an external organization. 

Table 3.1 Number of classrooms per grade 

Grade 

Number of Spanish 

Number of mainstream 

Total number 

immersion classrooms 

classrooms 

of classrooms 

1 

1 

2 

3 

PK 

K 

First 

Second 

 

6 

4 

5 

6 

5 

3 

3 

3 

22 

Table 3.1 (cont’d) 

Third 

Fourth 

Total 

 

 

2 

2 

11 

2 

3 

19 

4 

5 

30 

The students. My study took place in one of the third-grade Spanish immersion 

classrooms, during the 2017-2018 academic year. Initially, there were 21 students: 15 female 

students (one Latina, one Asian American, two African American, 11 White) and six male 

students (two Latino, one African American, three White). One student moved to a mainstream 

classroom in the middle of the year. All students spoke English at home primarily, including the 

three Latino/a students whose use of Spanish at home was limited. Most students (18) attended 

the language immersion program since either PK or kindergarten, and the other three students 

started in first grade. All students seemed comfortable sustaining conversations, as well as 

following directions and discussions in Spanish. Three of the students were on Individualized 

Educational Plans (IEPs). All three of them were pulled-out of mathematics class three times a 

week to work with the immersion program support aide in Spanish. 

The teacher. The teacher, Valery Abad, is a US-born Latina who considers both English 

and Spanish as her native languages. Growing up in Florida, she spoke Spanish at home 

exclusively with her Cuban parents and siblings. She holds a bachelor’s degree in bilingual 

Spanish education and a master’s degree in education, both from liberal arts colleges in the same 

city where the school is located. At the time of this study, she was in her fifth year teaching, and 

all her teaching experience had been in the third-grade Spanish-immersion classroom at this 

 

23 

school. She was actively involved in school activities and frequently volunteered to support 

various programs and events and to serve as mentor teacher to student teachers. 

Researcher role and positionality. Acknowledging that students can influence the 

culture of participation if teachers are willing to carefully consider their perspectives meant that 

rather than approaching this site as a knowledge-bearer, I joined the students and the teacher in 

ongoing dialogue about their participation. I assumed the role of a participant-observer (Glesne, 

2011) working with the students and the teacher with a genuine interest in understanding their 

participation-related perspectives and practices. The synergy between Valery’s desire to learn 

with her students and my commitment to “openness, curiosity, desire and willingness to interact 

in collaborative ways” (Glesne, 2011, pp. 157-158) invigorated our interest in this collaboration. 

Our common ground as Latina/o, bilingual educators facilitated rapport building, and our 

differences—Valery as a Cuban-America teacher and I as a Colombian international graduate 

student—enriched our collaboration. 

I made sense of the culture of participation in this classroom, while being aware of my 

influence in this context, as I was moving along the insider-outsider researcher positionality 

continuum (Foote & Bartell, 2011; Glesne, 2011). Over time, I became familiar with classroom 

norms, participated in classroom routines and, in some ways, students interacted with me as an 

insider to this classroom. Together, the students, the teacher, and I gradually developed trust that 

facilitated the dialogues we had about participation and about the connections between reflection 

and practice. Working together with the students and the teacher in their mathematics lessons, I 

adopted practices related to the culture of participation of this classroom and I needed to reflect 

on my involvement in these practices to bring to my awareness my interactive role in the 

evolving culture of participation. 

 

24 

The types of relationships we developed in this classroom along with my dissertation’s 

focus on joining students in reinterpreting what it means to participate required challenging 

hierarchies that favor researchers’ or teachers’ knowledge. Accordingly, the methodological 

decisions I made included using a participatory design research with children methodology to 

study a classroom issue that was meaningful to the teacher, the students, and the researcher. The 

issue of transforming the culture of participation emerged through several conversations and 

after two years preparing the collaboration, which I discuss in the following section. 

Preparing the Research Collaboration 

My initial research on this site focused on the interplay between mathematical activity 

and language use in classrooms where the language of instruction is different from students’ first 

language. In the spring of 2016, I observed two mathematics units, and Valery and I co-analyzed 

teaching episodes. In the fall of 2016, I observed Valery’s new third-grade math class, and she 

and I discussed possible issues for further research. Valery got interested in whole class 

discussions, noting that students tended to focus on giving correct answers. She envisioned 

students explaining their ideas and addressing each other. I shared in this interest and we started 

studying and incorporating number talks (Parrish, 2010) in her lessons. Initially, we decided 

together on a topic and I facilitated the number talks. We took this opportunity to explicitly 

discuss with children practices such as explaining their strategies, listening to each other’s ideas, 

and justifying reasons to agree or disagree. After I conducted three number talks, Valery began to 

select and conduct number talks on her own. 

Once students seemed more comfortable talking about their mathematical ideas, Valery 

and I focused on designing tasks and orchestrating discussions that would support the types of 

interactions Valery envisioned. We read and discussed articles from practitioner-oriented journals 

 

25 

(Breyfogle & Williams, 2008; M. S. Smith, Hughes, Engle, & Stein, 2009), and we studied 

groupworthy tasks (Featherstone et al., 2011). From November 2016 to June 2017, we designed 

and implemented tasks that had multiple possible solutions and multiple possible strategies, and 

that asked students to work collaboratively and explain their ideas. For each task, we anticipated 

a few strategies that were likely to emerge during small group work. We planned open-ended 

questions to probe and extend student thinking. Either one of us led the launching of each task 

and the whole class discussion that followed. We both supported student work in small groups. 

During the fall of 2017, I observed math lessons in Valery’s new third-grade group. I also 

occasionally observed recess, science, social studies, and the end of day reflection. For the 

mathematics class, the teacher and I continued using number talks and groupworthy tasks that I 

either observed or co-taught. Over time, Valery thought that although students seemed 

comfortable engaging in these tasks and discussing mathematical ideas, there were noticeable 

differences in participation. She was concerned with the pattern of a few students tending to 

dominate discussions and others tending to be quieter or limit their spoken contributions to 

agreeing or disagreeing with their peers. In my conversations with students, the notion of 

participation as involving more than talk emerged, which motivated my interest in bringing the 

students’ and Valery’s perspectives into dialogue. 

Methodology 

In the process of developing a methodology that would cohere with the participatory 

design of my dissertation, I recognized the key role of the students, the teacher and the researcher 

coming together to reinterpret what it meant to participate in this classroom. At the same time, I 

was also aware that translating these emerging reinterpretations of participation into teaching 

practice would involve rethinking the role of the teacher and ongoing reflection, and 

 

26 

reconsidering the types of lessons and mathematical activities, as discussed in the theoretical 

framework. Accordingly, I wanted to develop synergies between two related methodological 

processes: the process of data generation and analysis and the collaborative process of fostering 

transformations in the culture of participation. Data generation and analysis needed to 

simultaneously provide evidence of the evolving culture of participation, orchestrate 

opportunities for the teacher to question preconceived ideas of what it means to participate, and 

support her in revising tasks, lessons and teaching strategies. For this purpose, I used two 

complementary methodologies: participatory design research (PDR) and research with children. I 

draw on PDR because of its simultaneous focus on structuring collaboration and learning from 

the collaboration. I draw on research with children because of its attention to children’s 

perspectives that are foundational to the design of a collaborative research. The resulting 

methodology, participatory design research with children, is consistent with my 

conceptualization of participation as a phenomenon influenced by all involved and it supported 

the teacher’s process of learning to listen hermeneutically for participation. 

Participatory design research. Participatory design research is consistent with this 

study’s interest in maintaining partnerships that facilitate collaborative design, implementation, 

and analysis of interventions that address local (i.e. classroom) issues (Bang & Vossoughi, 

2016). Design based research—one of the frameworks that PDR integrates—defines 

interventions as a set of commitments and actions to address a classroom issue (Bell, 2004; 

Engeström, 2011). Design based research involves cycles of planning, implementing and 

reflecting interventions based both on theory and on teachers’ perceptions of the issue of interest. 

In this dissertation, this methodology involved iterative cycles of designing, analyzing and 

refining interventions to transform the culture of participation. The overall purpose of these 

 

27 

cycles of intervention was to provide recurrent spaces for all participants—teacher, students, and 

researcher—to engage in hermeneutic listening. By coming back to these spaces, each time with 

broader, more complex ways of listening and seeing, the teacher and I expected to provoke a 

departure from listening for predetermined forms of participation toward acknowledging and 

fostering ways of participating previously ignored. 

Participatory research—the second framework that PDR integrates—seeks to challenge 

power hierarchies, including the positioning of the researcher as an expert that comes to the site 

to execute a predetermined agenda (Cornwall & Jewkes, 1995; Fals-Borda, 1987). Participatory 

approaches do not regard the researcher as the producer of knowledge, and the researched as the 

object of analysis and the recipient of knowledge. Instead, participatory researchers move from 

working on or for participants to working with them (Berg, 2004; Bergold & Thomas, 2012; 

Hansen, Ramstead, Richer, Smith, & Stratton, 2001; Powell, Jeffries, & Selby, 1989). 

Participatory methodologies are consistent with this dissertation’s theoretical framework that 

focuses on the teacher reinterpreting with students what it means to participate in this classroom. 

In PDR, the process of partnering—establishing, maintaining and reflecting on the 

partnership—is foundational to the research (Bang & Vossoughi, 2016; Gutierrez, Engeström, & 

Sannino, 2016; Vakil & de Royston, 2016). This process contributes to making interventions 

likely to be legitimate and relevant to participants (Ehret & Hollett, 2016; Gutierrez et al., 2016). 

In this dissertation, the design, implementation, and analysis of the interventions did not depend 

solely on the researcher. Instead, Valery and I collaborated in these iterative processes, drawing 

on students’ perspectives. This approach contributed to the robustness of interventions—by 

including the perspectives of those most directly affected—and the robustness of the analysis, as 

we co-developed different insights and interpretations. The PDR approach also provided 

 

28 

opportunities for the teacher to engage in hermeneutic listening for participation. Each 

intervention cycle involved reflecting on taken for granted ideas about participation and about 

the teacher’s role on participation, as well as engaging in participatory interactions where the 

students influenced the unfolding of the lessons. Teaching strategies associated with hermeneutic 

listening, such as asking genuine questions, directing students’ listening and amplifying students’ 

participation, informed the design and implementation of interventions. 

Research on mathematics education that uses PDR has focused primarily on issues 

involving adults (Steinberg & Kincheloe, 2010). Examples include parents, teachers and 

researchers collaborating around issues of reform-based mathematics teaching (Booker & 

Goldman, 2016) and the collaboration between an instructor and his adult students to transform 

an undergraduate class pedagogy (Powell et al., 1989). Given that this dissertation involves 

children, in the following section I synthesize concepts from research with children that inform 

my study. 

Research with children. Research with children methodologies recognize that children 

are knowledgeable about their own experiences and that they can play a role in transforming 

their circumstances. Doing research for or on children frequently assumes that children are 

empty vessels or unfinished individuals in transit to adulthood (Clark & Moss, 2011; Cook-

Sather, 2002; Groundwater-Smith et al., 2014; Punch, 2002). Instead, doing research with 

children capitalizes on the belief that children act according to and have insight into the cultures 

of their age groups (Kirk, 2007; Rudduck, 2007). In this dissertation, the design and analyses of 

interventions involved children’s perspectives, as a methodological innovation for revising 

current views on the issue of mathematics class participation. 

 

29 

This dissertation embraces methodological commitments that are sensitive to children’s 

expertise and needs, including being flexible (Clark & Moss, 2011; Jacquez, Vaughn, & Wagner, 

2013), suspending control at different stages of the process (Boyden & Ennew, 1997; Jacquez et 

al., 2013; N. Thomas & O'Kane, 1998), and attending to participants’ perspectives (Kirk, 2007; 

Punch, 2002). These commitments echo tenets of hermeneutic listening that focus on 

participatory interactions that foster mutual influence, as teachers learn with students to broaden 

the ways they see, hear, and feel participation. A way to share control and attend to children’s 

perspectives is to flexibly draw on multiple methods that are appropriate for particular children 

(Hill, 1997; Kirk, 2007; Punch, 2002). Children’s comfort participating in methods traditionally 

used with adults varies depending on the topic of discussion, their age, the researcher-

interviewee relationship, self-confidence, and language skills (Punch, 2002). Accordingly, I drew 

on multiple data generation methods, as I describe next. 

Data Sources 

There were two stages of data generation. The first stage focused on supporting the 

teacher to bring into dialogue the students’ and her initial perspectives on what it means to 

participate. The second stage focused on connecting the teacher’s reflection to her practice. 

Reflections on what it means to participate. Bringing the students’ and the teacher’s 

perspectives on participation into dialogue to begin to reinterpret participation involved eliciting 

students’ perspectives. I conducted multi-methods focus groups with students to discuss 

participation in general and in the mathematics class specifically (see Appendix A). The structure 

of focus groups resonated with the consideration of both the individual and the social influencing 

participation, as individual contributions frequently prompted conversations where the group co-

 

30 

developed ideas. Additionally, working with small groups and using multiple methods helped me 

attend to all students, as opposed to the most outspoken only. 

I conducted six focus groups with three or four students in each group, during the second 

week of April 2018. Consistent with tenets from research with children methodologies, the 

structure of these focus groups was flexible. I continuously reiterated that participation was 

voluntary and that students could choose whether to join a focus group or not. I informed 

students that I was going to share their ideas with the teacher. Students chose their own groups, 

an important consideration to help the children feel comfortable sharing their ideas. Students 

could also choose to leave the group at any moment and join another group later, or not join at 

all. Two students took up the option of leaving a group and came back a different day. Although I 

started the groups in Spanish, I asked students to express their ideas in either Spanish or English. 

In most groups we ended up using both languages at different times. 

Students could also choose among multiple methods to share their ideas. I initially asked 

students to think about what it meant to participate to them and how they participated in the 

mathematics class. Each student individually decided whether to draw or write about these ideas 

or to think and then share during the conversation. Paper and markers were available and all but 

one student used these resources. After this individual part, each student could decide whether to 

share ideas with the rest of the group or to share privately with me after the focus group. All 

students shared their drawings and writings and explained what they represented. I asked follow 

up questions to clarify or to ask for specific examples. In all groups, students frequently 

addressed each other, adding to what others said or proposing alternative ideas. Each focus group 

lasted about 30 minutes. I audio recorded and transcribed each group and I collected students’ 

work. 

 

31 

Acknowledging the teacher’s role as a full participant in classroom activity and as an 

insider on the culture of the classroom, I also conducted an interview with Valery (see Appendix 

B). This 45-minute semi-structured interview took place after all the focus groups, and the 

teacher and I discussed her ideas about participation in her mathematics class. We also discussed 

context-specific situations that she thought influenced her perception about participation and 

how she addressed participation-related issues. I audio recorded and transcribed the interview. 

Recognizing my role as a participant-observer directly engaged in the classroom culture of 

participation, I wrote an analytical narrative reflecting on my own perspective on participation in 

general and in this mathematics classroom. 

After I conducted an initial analysis of the focus groups and teacher interview data (as 

described in the Data Analysis section below) Valery and I co-analyzed some of the responses to 

bring into dialogue the students’ and the teacher’s perspectives on participation. This was a 

reflection opportunity where the teacher began to expand what she considered participation. I 

took field notes during this conversation. Valery and I also discussed how these emerging 

reflections of participation could inform the design and implementation of intervention cycles, 

which I discuss next. 

Connections between reflections on participation and practice. Between April and 

June of 2018, the teacher and I engaged in the intervention cycles. Informed by the PDR 

methodology, the cycles included co-designing, implementing and reflecting on interventions. In 

the co-designing phase that was informed by the teacher’s reflection, we engaged in an iterative 

process of planning and revising teaching plans. As the teacher implemented and reflected on 

these plans, we revised subsequent ones in response to the directions each unit took. I took field 

notes documenting the ideas from the first stage of data generation that guided each intervention, 

 

32 

the practitioner-oriented articles we read, and the ideas for lessons that we discussed. These 

designs were flexible and we made adjustments based on unanticipated ways of participating and 

mathematical ideas emerging in interaction with the students. 

During the implementation phase, I video recorded 14 mathematics lessons (two lessons 

per week, for seven weeks), using two cameras. One was a handheld camera following the 

teacher and connected to the lapel microphone she wore. The other one was a wide-angle camera 

set on a tripod recording the entire classroom. I set an audio recorder at one of the students’ 

tables to capture audio from different parts of the classroom. This set up helped see and hear both 

the teacher and the students. I used the Final Cut Pro software to create a unified video for each 

lesson, with a split screen showing the recording from both cameras. I transcribed all videos 

within 24-hours of the recording. Transcripts included spoken utterances in their original 

language, as well as descriptions of gaze, gestures, and movement related to the exploration of 

mathematical ideas. I also wrote analytic field notes (Glesne, 2011) to document tentative 

preliminary interpretations and questions about the transformation of the culture of participation. 

The teacher and I reflected on the interventions and discussed adjustments to lessons, 

possible teaching strategies to try, and questions or insights about participation. I also had 

conversations with students whose participation continued to puzzle the teacher and me. In these 

conversations, I asked specific students about their participation, including what they were doing 

during a task, and how other students, the teacher and I helped them participate, or come up with 

or understand an idea. I had six of these conversations with different students, and I started each 

conversation letting the student know I had questions for my research, and that I would share 

notes from our conversation with the teacher. I took field notes about these conversations. In 

addition to these ongoing conversations both with the teacher and the students, during the 

 

33 

summer of 2018, the teacher and I discussed how the dynamic connection between reflection and 

practice facilitated a process of reinterpreting what it meant to participate. We conducted four 45-

minute sessions co-analyzing data (as described below). I audio recorded and transcribed these 

sessions. 

This methodological design of the PDR with children generated multiple sources of data 

that elicited the students’, the teacher’s and my participation-related perspectives. These sources 

of data are also consistent with the focus of hermeneutic listening on participatory interactions to 

transform classroom experience. Additionally, the co-analysis of interventions was part of the 

process of learning to listen hermeneutically for diverse student participation. 

Social Semiotics Analysis of Connections Between Reflection and Practice 

I draw on social semiotics (Halliday, 1978; van Leeuwen, 2005) as an analytical 

framework to interpret the connections between the teacher’s reflection and practice because 

understanding these connections is key to understanding the transformation of the culture of 

participation in the classroom. Acknowledging the influence of social contexts in meaning 

making, social semiotics focuses on the processes of mutual influence where culturally situated 

meanings emerge (Morgan, 2006; van Leeuwen, 2005). In this study, social semiotics guided the 

exploration of the mutual influence between the students’ and the teacher’s perspectives on 

participation that supported a transformation of what it means to participate in a mathematics 

classroom. 

Instead of assuming that pre-established static meanings reside in specific semiotic 

resources (i.e. resources for meaning-making), social semiotics is concerned with the way 

meanings emerge in interaction and how social contexts influence these meanings (Halliday, 

1978; van Leeuwen, 2005). Semiotic resources include language, images, symbols, gestures, and 

 

34 

artifacts that interact to support meaning-making (Vannini, 2007). I explore how the teacher 

learned to recognize and validate participation in students’ use of multiple semiotic resources that 

transcended spoken language. 

Social semiotics has been used in mathematics education research to make sense of 

multiple issues, including: how teachers and students continuously rearrange their relationships 

with each other, with mathematical ideas, and with multiple semiotic resources (Hoffmann, 

2006; O'Halloran, 2005; Sáenz-Ludlow & Presmeg, 2006); how teachers’ perceptions of 

students’ socioeconomic backgrounds influenced classroom discourse (Atweh, Bleicher, Cooper, 

1998, 1998); or how social factors influenced secondary students’ construction of the nature of 

school mathematical activity (Morgan, 2006). Previous research has also explored how social 

semiotics can help teachers become aware of dominant discursive practices (de Freitas & 

Zolkower, 2009; 2011) or of the value that students see in using specific semiotic resources 

(Hoffman, 2006). In this dissertation, I draw on social semiotics to study the connections 

between reflection and practice, a process that began by bringing into dialogue the students’ and 

the teacher’s perspectives on participation, which I explain in the following section. 

Different meanings of participation. In spite of the relative stability and cohesion that 

characterizes cultural groups, they are not monolithic and the meanings and values that different 

members attribute to specific practices differ. In social semiotics, connotations are variations 

within the culturally shared meanings, values, and practices of a social group. Frequently, 

children’s connotations of specific aspects of a culture are ignored or suppressed (van Leeuwen, 

2005, p. 22). Counter to this social practice, in this dissertation it was important to elicit what it 

meant to participate for both the students and the teacher. The first stage of data generation 

 

35 

focused on eliciting these meanings, to provide material for the teacher’s reflections on 

participation. 

Understanding transformations in the culture of participation. Over time, certain 

practices and meanings associated to cultural phenomena tend to go unnoticed and unquestioned. 

Semiotic innovation occurs when specific practices and resources take on new meanings in a 

social group (van Leeuwen, 2005). Members of the group may not put in words the 

transformations of the practices, but they recognize the novelty and respond to it in different 

ways, from adoption to resistance (van Leeuwen, 2005). I draw on semiotic innovation to 

explore the evolving transformations in the culture of participation that emerged not from the 

teacher’s reflections or her practice but from the connections between reflection and practice. 

Semiotic innovation occurs when different processes interrupt cultural traditions, static 

roles and overreliance on one semiotic resource that contribute to the stagnation of meanings 

associated to a cultural phenomenon (van Leeuwen, 2005). For example, teachers frequently 

assume that their role is to ensure the observance of norms that traditionally regulate classroom 

interactions, such as having students raise their hand to get a turn to talk in a class discussion. 

Semiotic innovation can occur when the teacher rearranges the way that students socialize ideas, 

thinking of alternatives to traditional discussions where teachers assume the role of moderators 

relying primarily on spoken language. In this dissertation, the second stage of data generation—

the design and implementation of cycles of interventions—sought to provoke the occurrence of 

semiotic innovation. 

I focus on the interaction among challenging the authority of traditions, role shifting, and 

reassigning saliency—all of which contribute to semiotic innovation (van Leeuwen, 2005)—to 

explore transformations in the culture of participation that developed as the teacher connected 

 

36 

reflection and practice. Engaging in the hermeneutic listening process of interrogating taken for 

granted perspectives on participation involves challenging the authority of traditions that prevent 

teachers from thinking differently about participation, and assuming the role of a participant in 

mathematical activity as a listener that makes room for multiple ways of participating. Teaching 

strategies inspired by hermeneutic listening, including attending to more than spoken language 

and amplifying students’ participation, helped the teacher reassign saliency to the use of multiple 

semiotic resources she was learning to see as legitimate participation. 

Collaborative Data Analysis 

Collaborative data analysis served to expand opportunities for the teacher to reflect on 

participation and to connect this reflection to practice. In the following section, I discuss the two 

stages of data analysis in which the teacher and I collaborated. 

Analysis of perspectives on participation. This stage of data analysis focused on what it 

meant to participate for the students and for the teacher, as well as on bringing these different 

perspectives into dialogue. I did an initial transcript analysis of the student focus groups and the 

teacher interview. I annotated transcripts focusing on nuances in characterizations of what it 

means to participate. My experience as a participant observer in this classroom allowed me to 

make sense of students’ responses and to follow up with them when necessary. Some students’ 

statements were salient because they clearly represented my perception of the student’s usual 

ways of participating, or because they were unexpected and surprising. For example, some 

outspoken students made comments problematizing whole class discussions. These comments 

were salient and intriguing for me because they revealed unknown participation related issues 

that even students who were apparently comfortable and skillful at navigating discussions 

experienced. I also annotated statements where students agreed, disagreed or built on others’ 

 

37 

responses during a focus group, identifying convergence and divergence of perspectives on 

participation among students. I analyzed annotations across transcripts to identify recurrent 

ideas, which I interpreted as indicating relatively shared values about specific ways of 

participating. I contrasted recurrent ideas in the students’ and the teacher’s transcripts to identify 

divergence in the ways of participation that they valued. 

Consistent with the collaborative nature of this dissertation, my initial transcript analysis 

was tentative and exploratory and the teacher and I engaged in co-analysis. This collaborative 

data analyses served as a reflection opportunity to carefully consider students’ perspectives on 

participation. I selected vignettes of responses from the focus groups and from the teacher 

interview to co-analyze. Some vignettes illustrated the contrasting ideas that the students and the 

teacher had about participation. Other vigenttes included ideas that were unique and intriguing, 

having the potential of helping us make sense of participation in more inclusive ways. For 

example, while the idea of helping others as a legitimate way of participating was recurrent in 

the focus groups, only one student mentioned asking for help. Considering both helping and 

asking for help as complementary ways of participating eventually became the focus of one of 

the intervention cycles that I describe in Chapter 5. 

In the co-analysis of these vignettes, I asked the teacher to relate the students’ 

perspectives to her own perspectives on participation. I shared my initial annotations on the 

vignettes and frequently refined my tentative interpretations. During this co-analysis, the teacher 

and I began to discuss how to address ideas from this emerging reflection on participation in the 

intervention cycles. As these ideas evolved in the iterative process of designing and revising 

interventions, I analyzed the connections that the teacher drew among these initial ideas, her 

ongoing reflections on participation and her practice. 

 

38 

Analysis of connections between reflection and practice. The second stage of data 

analysis took place in the summer following the school year and focused on video data from the 

intervention cycles. Similar to the first stage, I conducted an initial analysis of the entire video 

data. I focused on interactions that illustrated evolving reinterpretations and legitimization of 

what it meant to participate in this classroom. I drew on my knowledge of the culture of the 

classroom to identify these interactions. For example, I noticed that when students would ask a 

question about a topic that was not included in the third-grade pacing guides, the teacher would 

explain that the topic was beyond their grade level. During the implementation phase of this 

study, however, the teacher began exploring some of these questions, as I describe in Chapter 5. I 

also drew on the students’ and the teacher’s comments regarding how they were feeling that 

something new or unusual was happening. That was the case when students asked to engage in 

one of the co-designed teaching strategies that Valery implemented, or when the teacher shared 

her excitement about students’ positive response to a way of participation that was the focus of 

an intervention cycle and that the class had not considered before. 

After identifying interactions that illustrated transformations of the cuture of 

participation, I preselected three videos (one from each intervention cycle) that I considered 

included pivotal moments in the transformations of the culture of participation as candidates to 

co-analyze with the teacher. Valery and I discussed these candidates and changed one of the 

videos for another one she was particularly interested in because of its focus on a teaching 

strategy that she was curious about. 

From each of the videos, we watched excerpts that I had annotated. We focused on how 

the teacher recognized and drew attention to multiple ways of participating and how students 

responded and specific mathematical ideas emerged. We discussed our perceptions of 

 

39 

transformations in the ways of participating that influenced the unfolding of classroom 

interactions. Using these insights, I went back to the entire video data set to refine my 

interpretations. I attended to challenging the authority of tradition, role shifting and reassigning 

saliency to multisemiotic resources as contributing to semiotic innovations to promote the 

legitimization of multiple ways of participating. I also added notes and quotes from the co-

analyses sessions related to school influences on classroom participation, and notes from 

conversations with children during the interventions. Considering multiple data sources that 

elicited different participants’ perspectives strengthen the robustness of the interpretations. 

Finally, the teacher and I co-analyzed the connections between her reflections on 

participation and her practice. We focused on her learning to listen differently, more 

hermeneutically, and how this different way of listening allowed her to see participation more 

broadly, including her role in amplifying multiple ways of participating. This collaborative data 

analysis facilitated the emergence of factors that influenced the connections the teacher 

established between reflection and practice, including raising perspectives on participation to 

awareness, taking the role of a learner of student participation, and flexibly revising tasks, 

lessons and units that could support multiple ways of participation that students valued. 

Collaborative data analysis is also consistent with the participatory research framing of validity 

in terms of the collaboration’s usefulness for all participants (Bergold & Thomas, 2012) and its 

relevance to both research and practice (Creswell & Miller, 2000, p. 128). 

 

 

40 

Chapter 4 Broadening the Teacher’s and Researcher’s Perspectives on Participation 

Beginning in this chapter and continuing in Chapters 5 and 6, I present findings that 

illustrate the interactive and evolving process of reflecting, preparing to, and engaging in 

transformative practice. In this chapter I describe how bringing the students’ and the teachers’ 

perspectives on participation into dialogue helped the teacher broaden her ideas on what it meant 

to participate. In Chapter 5 I describe how reflections informed the teacher-researcher co-design 

of interventions, preparing the teacher to promote diverse ways of participating and to address 

issues that stifled participation. In Chapter 6 I describe ongoing connections the teacher 

developed between reflection and practice, as she engaged in hermeneutic listening for 

participation that fostered transformations in the culture of participation. 

I begin by discussing how opening up to alternative ways of participating involved letting 

go of deep-seated ideas regarding what counts as participation. Understanding nuances and 

diversity within the situated, context-specific culture of participation laid the ground for the 

teacher to begin engaging in hermeneutic listening for participation. In this reflection phase, 

Valery began to broaden her perspective on participation by learning to see beyond quantity and 

instead focusing on qualities of participation. She also began to expand her ideas by learning to 

locate participation in time and space, thus allowing herself to consider the lasting effects of 

participation and the students’ multiple ways of enacting participation in the classroom. 

Unanticipated student responses piqued her curiosity, thus confronting her with tensions that 

motivated her to reimage her practice. All transcripts are presented in their original language, 

with English translation provided in italics. 

 

 

 

41 

Learning to See Participation as Qualifiable 

Valery began to question her perspective on participation as primarily quantifiable by 

considering her students’ perceptions as something qualifiable. I illustrate this connection with 

an episode from a student focus group where a binary of either presence or absence of 

participation came up and was challenged by the group (Table 4.1).

 

42 

Table 4.1 Example of participation as qualifiable and as quantifiable 

Student perspectives 

Teacher perspectives 

Paige: 

If you won't care and try to get the answers right and 

Valery: Los uso [los palitos] porque así [los estudiantes] saben 

read… and if you're just being lazy and cheating, 

que cualquiera puede participar, no sólo los que siempre 

that's not participating… You need to read the 

levantan la mano… No siempre funciona porque pueden 

problems and try. Then you're participating. 

pasar si no quieren participar, y entonces yo escojo otro 

Willie: 

Yo no estoy de acuerdo con este porque yo no puedo 

nombre, pero me gustaría ver que todos participen… 

estar aquí y no participar. Si yo estoy aquí, estoy 

Porque a veces algunos no participan. Y hay varias 

participando. 

formas de participar, no tiene que ser todos igual, pero 

 

I don’t agree with that because I cannot be here and 

que todos participen. 

not participate. If I’m here, I’m participating. 

I use them [the sticks] because then they [the students] 

Mackenzie: Mío es lo mismo. Yo no intentó escribir qué es 

know anybody can participate, not only those who 

participar y yo vio el papel de Paige, pero yo estoy 

always raise their hand. It doesn’t always work because 

participando porque yo estoy aquí… Yo estoy 

they can pass if they don’t want to participate, and then I 

escuchando y estoy viendo ahora. 

draw another name, but I would like to see everybody 

Same here. I didn’t try to write what it is to  

participating… Because sometimes some [students] don’t  

43 

 

 

 

Table 4.1 (cont’d) 

participate, and I did look at Paige’s paper, but I’m 

participate. And there are different ways of 

participating because I’m here… I’m listening and 

participating, it doesn’t have to be the same way for 

I’m looking. 

everyone, but [what’s important is] that everyone 

participates. 

 

 

44 

In the student responses, Paige depicts a binary of participation and non-participation and 

she associates specific behaviors with either category in the binary. In contrast, for Willie and 

Mackenzie their presence in the classroom constitutes a valid form of participation. As described 

before, there were two stages in the focus groups: in the first stage, students chose whether to 

write, draw or think about participation, and in the second stage they shared their ideas. In the 

excerpt, Makenzie explained that in the first stage she did not write what it meant to participate 

and she looked at Paige’s paper, which are behaviors that Paige associated with non-

participation. Mackenzie, however, argues that she was participating. Her comments suggest that 

rather than identifying participation or non-participation in specific moments (i.e. in the first or 

in the second stage of the focus group), for her participation occurred over the course of an entire 

meaningful task. Instead of thinking that she only participated during the second stage but not in 

the first stage of the focus group, Mackenzie preferred to locate her participation in the whole 

focus group in ways that included the hermeneutic abilities of listening and observing. 

Valery’s characterization of participation was closer to the absence or presence binary 

that Paige mentioned. For Valery, the presence of participation—students sharing an idea or 

answering a question—implied a simultaneous absence of participation in that not every student 

could actually participate during whole class discussions. When she called on a student, the 

student chose between participation and non-participation by either accepting or declining 

Valery’s invitation to talk. Since students’ response to her invitation could vary every time, for 

Valery the status of participation or non-participation was not static but changeable over time. 

Yet, unlike Mackenzie who considered listening and observing as legitimate ways of 

participating, Valery’s perspective was grounded primarily in student talk. 

 

45 

Considering these contrasting connotations of participation, Valery began to broaden her 

perspective on participation as she began to consider the situated ways in which participation 

occurred in her classroom. She experienced a tension between her curiosity about students’ 

responses and apprehension about the implication of these responses on her teaching. 

Reinterpreting non-spoken participation as legitimate participation, Valery questioned her goal of 

trying to distribute talk time uniformly among students. This, however, conflicted with her 

commitment to preventing the monopolization of discussions by a few students. Navigating this 

tension involved reconsidering the relationship between participation and talk, which I illustrate 

in the contrasting student and teacher comments in Table 4.2.

 

46 

Table 4.2 Examples of problematizing participation as talk 

Student perspectives 

Teacher perspectives 

Calum: Tú puedes participar [cuando] muestras a todos qué hiciste 

Valery:  Casi siempre hay alguien que quiere participar, 

y explicas tu estrategia. Pero si no escuchan… o si no te 

alguien que levanta la mano para explicar algo o 

entienden o si Señora Abad es como ‘oh sí. Siguiente,’ 

alguien que quiere mostrar qué hicieron. Es pocas 

entonces es como, ¿para qué hablaste? 

veces que pasa que nadie quiere [participar], 

 

You can participate [when] you show everybody what you 

entonces hay que darles más tiempo y tengo que 

did and explain your strategy. But if they don’t listen… or if 

pensar si están distraidos o a lo mejor están 

they don’t understand or if Mrs. Abad is like ‘oh yes. Next,’ 

confundidos. 

then it’s like, why speaking out? 

 

There’s almost always someone who wants to 

Lia:  Puedes participar si dices la respuesta. Pero esto es posible 

participate, someone raising their hand to explain 

no deja que todos piensan porque ya saben la respuesta. O 

something or someone who wants to show what they 

es posible tú no sabe por qué es la respuesta. Entonces 

did. There are few times when no one wants to 

puedes participar pero es un poquito mal participación. Es 

[participate], then I have to give them more time and 

buen participación si tú explica la respuesta o si tú espera 

I have to think whether they are distracted or maybe 

que todos piensan. 

 

they are confused. 

 

 

47 

Table 4.2 (cont’d) 

You can participate if you tell the answer. But maybe this 

 

doesn’t let others think because they already know the 

answer. Or maybe you don’t know why that is the answer. 

So, you can participate but it’s a little bit bad participation. 

It’s good participation if you explain the answer or if you 

wait for everybody to think. 

 

 

 

 

48 

Students’ perspectives on participation transcended talk and included the effect of such 

talk. Calum’s comments problematized the tendency of equating talk with participation as, for 

him, talking did not necessarily mean desirable participation. Instead, he qualified participation 

in relation to the effects that his talk could have on others’ thinking. Talk that did not influence 

others’ thinking or the direction of a lesson was pointless participation for Calum. Lia’s 

comments added complexity to the relationship between talk and participation. Rather than 

valuing talk in itself as a legitimate way of participating, how she qualified such participation 

depended on whether talk supported or hindered other people’s participation. She problematized 

talk that denied others the opportunity to think or that did not illuminate how to develop a 

mathematical idea. Both Calum and Lia valued participation that positively influenced others. 

In contrast with students’ perspectives on participation as influencing others, Valery’s 

perspective suggests a concern in the delay of talk rather than the effect of talk. While Lia related 

time to think to productive participation for all, Valery considered time to think as absence of 

participation associated with misbehavior or lack of understanding. This suggests that Valery 

perceived participation as immediate responses to her questions, regardless of if and how the 

class took up those responses. 

As the teacher expanded her perspective to include qualifying participation, she was well 

on her way to rethinking what counted as participation. Initially, she showed some resistance to 

the idea of participation as being present without speaking out, focusing on what she perceived 

as a tension between acknowledging non-spoken participation and disrupting the monopolization 

of class discussions. Recognizing that listening and observing could be valid ways of 

participating and that uniformly distributed talk could be experienced as irrelevant or 

counterproductive, Valery was left with many questions that needed to be addressed. She 

 

49 

wondered about how she could bring all students’ ideas into meaningful dialogue where they 

could influence each other’s thinking in the limited time she had for each lesson. She also 

wondered about what else students saw as enhancing or stifling participation that she had not 

considered before. 

I interpreted these wonderments as a productive destabilization of Valery’s perspective 

on participation that could inform her hermeneutic listening for student participation. Rather than 

reducing participation to student talk, Valery began to question what, besides spoken language, 

she could interpret as participation, and how she could engage with students in a participatory 

interaction to influence the unfolding of lessons. These wonderments motivated Valery to 

consider how she could join the students in connecting ways of participating such as listening, 

observing and thinking with the idea of mutual influence. 

Relocating Participation as a Dynamic Interaction Between the Individual and the Social 

While Valery’s initial interpretation of participation foregrounded individual student 

characteristics and preferences, she began to consider the role that social interactions played in 

how students experienced participation. Table 4.3 shows perspectives on participation that relate 

the social and individual as equally important aspects.

 

50 

Table 4.3 Example of over-competitiveness in whole class discussions 

Student perspectives 

Teacher perspectives 

Emma: …si tú eres como ‘yo, yo, yo’ [levantando la mano], 

Valery:  Algunos aprenden mejor hablando sobre lo que están 

entonces, Señora Abad va a call on you y puedes aprender 

pensando, entonces a veces casi que es inmediato 

más porque puedes explicar tus ideas. 

que termino una explicación y van a empezar a 

 

…if you are like ‘me, me, me’[raising her hand], then, Mrs. 

participar. Pero hay otros que no aprenden así y tal 

Abad will call on you and you can learn more because you 

vez participan sólo cuando yo escojo quién responde 

can explain your ideas. 

una pregunta o resuelve un problema, y ni así… Yo 

Calum:  Tienes que ser rápido. Levantas tu mano rápido para que 

sé que a veces Daniel va a estar haciendo otra cosa y 

Señora Abad te vea primero y entonces tú dices [tu idea] 

entonces luego no va a poder decir nada cuando le 

primero. Porque si no haces esto, entonces ella no te va a ver 

pregunte algo. O Emi… A ella no le gusta pasar a la 

o ella va a elegir a otra persona. Esto es bueno porque tú 

pizarra y contarnos sus estrategias. 

puedes escuchar a un otra persona pero entonces tú no 

 

Some learn better by talking about what they are 

puedes decir tu idea. También a veces, si muchas personas 

thinking, so sometimes it’s almost immediately after I 

dicen antes que tú, entonces cuando tú dices, ya no va a ser 

finish an explanation and they’ll start participating. 

nuevo o como interesante. 

But there are others who don’t learn like that and  

 

 

51 

Table 4.3 (cont’d) 

You have to be fast. Raise your hand fast, so that Mrs. Abad 

maybe they participate only when I pick who gets to 

sees you first and then you can tell [your idea] first. Because 

answer a question or solve a problem, and not even 

if you don’t do that, then she won’t see you and she will pick 

then… I know that sometimes Daniel will be doing 

someone else. That’s good because you can listen to 

something else and then he won’t be able to say 

someone else but then you cannot tell your idea. Also 

anything when I ask him a question. Or Emi… She 

sometimes, if many people tell [their idea] before you do, 

doesn’t like going to the board and telling us about 

then when you tell yours, it won’t be new or, like, interesting. 

her strategies. 

 

 

52 

Emma and Calum’s comments suggest that they experienced whole class discussions as 

over-competitive situations where students had to engage in a race to be the first one to capture 

the teacher’s attention. Rather than personal preferences or individual characteristics determining 

participation, a social arrangement of competitive interactions stifled their participation. Students 

declared how they strategized interactions in whole class discussions in order to secure 

participation that was influential to others. Students perceived that their participation could be 

inconsequential to others depending on when they got to share their ideas. For example, although 

Calum valued listening that came from others, for him listening meant not sharing his ideas and 

not influencing others’ thinking. During data analysis, Calum’s comments on whole class 

discussions were salient because he was a talkative student who seemed comfortable frequently 

speaking up in discussions. Learning how he perceived whole class discussions was revealing 

and drew my attention to the role that different types of interactions played in how students 

experienced participation. Other students shared similar strategizing to optimize their 

participation in whole class discussions, including raising their hand quietly to increase the 

chance of being called on or waiting to raise their hand until they had something original to 

share. While some students tried sharing early rather than later so classmates would not get bored 

and stopped paying attention, others tried sharing when the teacher was asking complex 

questions that allowed for generative answers. 

Unlike the students’ perspectives, Valery’s perspective on participation foregrounded 

individuality. Her comments suggest that rather than seeing interactions among students’ 

participation, she attributed participation and non-participation to students’ individual 

preferences. For her, these preferences were predictable, stable and unrelated to how others’ 

participated. Valery interpreted absence of student talk in whole class discussions as specific 

 

53 

students’ aversion to be on the spotlight, rather than related to an intimidating type of social 

interaction. She valued participation in the form of the quick responses from talkative students 

and she problematized what she perceived as the non-participation of quieter students. 

Analysis of these contrasting perspectives on participation in whole class discussions 

helped Valery to relocate participation into the students’ meaningful social interactions. Students’ 

comments surprised her, as she was more concerned with the different amount and frequency of 

student talk during discussions. Rather than discarding her knowledge about specific students’ 

preferences, she began to rethink participation as a dynamic interaction between the individual 

and the social, considering how the arrangement of social interactions influenced participation. 

She became increasingly interested in devising whole class discussions where students could 

experience a sense of mutual influence without feeling they had to compete for opportunities to 

participate in ways they liked. 

Relating individual and social influences on participation helped Valery reflect on social 

arrangements that could support participation. Table 4.4 shows perspectives that illustrated this 

interplay.

 

54 

Table 4.4 Example of participation as belonging to a group 

Student perspectives 

Teacher perspectives 

Marta:  I like it when we’re like [clutching hands]. So, we need to 

Valery:  Algunos son más líderes y pueden organizer el 

help each other and then we're all participating together and 

grupo y otros son más como de apoyar al grupo y 

doing stuff. Usually, that would help you participate because 

no toman la iniciativa… en las estaciones, a 

then you get lots of ideas or, like, cool stuff to do. 

veces tomo eso en cuenta para asignar roles o 

Coretta: When someone does everything and says like ‘OK, we have 

partes de la actividad y que luego junten sus 

to finish this page and I'm going to do this,’ and the rest of us 

partes, pero que yo pueda ver lo que cada uno 

only copy, I feel a little discouraged or almost as if the group 

sabe, lo que cada uno hizo. 

is scattered… so, we’re all participating but it feels like we’re 

 

Some are more leaders [than others] and they 

not in it together. 

can organize the group and others are more like 

Stacey:  I like helping others because I can help everyone feel part of 

they support the group but they don’t take 

the team. 

initiative… in the stations, sometimes I take that 

Jimmy: Sometimes you can participate too much because you’re not 

into account to assign roles or parts of the 

really helping but telling others ‘this is the answer.’  

activity and then they bring their parts together, 

Rose:  Yeah. You have to think and you have to listen first. 

but in a way that I can tell what each one knows,  

 

 

55 

Table 4.4 (cont’d) 

Jimmy:  And you can say like ‘I think we can try this way because I 

what each one did. 

draw it here and I think it works. What do you think?’ 

 

56 

Students’ perspectives suggest that they understood participation as membership to a 

social group. They valued participating in harmoniously coordinated activity characterized by 

collaboration, inclusion, and consideration of others’ ideas, which is in stark contrast with the 

over-competitiveness of discussions. Valery’s response suggests her concern with individuality, 

both in terms of which roles to assign to students depending on their individual characteristics 

and in terms of individual contributions to the task. 

Of the multiple ways of participating that Valery was learning to see, the idea of helping 

resonated with her earlier interest in encouraging students to help each other. This convergence 

between her students’ and her own perspectives served as an entry point for Valery to consider 

helping each other as a valid form of participation. She related students’ interest in helping to her 

interest in promoting whole class discussions that could amplify rather than limit participation. 

Some of her previous efforts to encourage helping frequently involved asking students who 

finished a task first to help classmates who needed help. Reflecting on this idea, Valery 

commented that requesting students to help others finish a task seemed too prescriptive and 

limited compared to the spontaneous and creative ways of helping that students described. 

Rethinking the Teacher’s Role in Participation 

Reflecting on the students’ and her perspectives on participation, Valery became aware of 

her perceptions on her role in student participation. While she initially saw her role as equitably 

eliciting student talk, she started to see her role as inviting students into meaningful mathematics 

tasks that acknowledged multiple ways of participating in cooperative interactions. The examples 

in Table 4.5 illustrate the contrast between students’ perception of how the types of tasks 

influenced their participation and Valery’s focus on eliciting student talk. 

 

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Table 4.5 Example of participation in different activities 

Student perspectives 

Teacher perspectives 

Stacey:  [Participate in] what? Like in charlas de números [number 

Valery: Tengo que estar mirando. No sólo en las discusiones 

talks], or when Señora Abad is explaining something, or 

sino también cuando están trabajando en sus grupos. 

when we work at our table? 

Si veo que alguien no está participando, a veces me 

Jimmy: It’s different but it’s the same. It’s the same because 

siento con ellos y le pido a alguien del grupo que no 

sometimes what we’re doing doesn’t matter. Like, I love to 

está participando que haga algo, como leer parte de 

ask questions all the time. 

un problema que les puede servir para pensar en una 

Juan:  Cuando no sabes la respuesta [a un problema] sí puedes 

estrategia. 

participar. Cuando no sabes la respuesta tú debes participar. 

 

I have to keep on watching. Not just during 

Tienes que hacer tu mejor. Tienes que preguntar y pedir 

discussions but also when they are working in 

ayuda, y escuchar y pensar. 

groups. If I see that someone is not participating, 

 

When you don’t know the answer [to a problem] you still 

sometimes I sit with them and I ask someone in the 

can participate. When you don’t know the answer you 

group who is not participating to do something, like 

should participate. You have to do your best. You have to 

reading part of a problem that can help them think of 

ask questions and ask for help, and listen, and think. 

a strategy. 

 

 

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Students’ perspectives suggest that how they participated varied according to the specific 

types of activities. For Stacey, my initial question about how they participated in mathematics 

class was too general, as she saw specific ways of participating according to the activities. 

Although Jimmy recognized that some ways of participating transcended the type of activity, 

student responses in different focus groups indicate they tended to associate talk with 

participation in whole class discussions. In Jimmy’s example, although asking questions may 

involve student talk, it is not a kind of talk where students explain or justify an idea but elicit 

others’ ideas and it may express curiosity, confusion, or the need for help. Juan’s response 

suggests that questions foster multiple ways of participating, including asking for help, thinking 

and listening. Although several students mentioned helping, Juan was the only student who 

mentioned asking for help. This idea was new to Valery and me and we eventually reconsidered 

the role of both helping and asking for help as a way of promoting collaboration among students, 

as I describe in Chapter 5. 

When discussing activities other than whole class discussions, students mentioned 

drawing, observing, writing, reading, and working with specific materials (e.g. rulers and base 

ten blocks) as valid ways of participating. Students’ comments did not seem to favor one way of 

participating over others or to subordinate one way to others. For example, instead of referring to 

listening or observing as simply facilitating talk, students described them as part of multiple, 

alternative and/or complementary ways of participating. During co-analysis, although Valery was 

intrigued by the idea of multiple ways of participating as alternative and complementary, she 

continued to see a tension between validating non-spoken participation and supporting students 

in talking about their ideas. She seemed to value non-spoken participation as long as it eventually 

translated into student talk, which initially obscured her ability to envision ways to recognize and 

 

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incorporate alternative ways of participating into lessons. This became a recurrent topic of 

discussion that we revisited and addressed in subsequent stages of the collaboration. 

Valery’s comments in Table 4.5 above suggest that she saw her role as interrupting what 

she saw as non-participation, rather than wondering how students participated in a specific 

activity and how to support or amplify their participation. Learning about students’ perspectives 

on how the types of activities influenced how they participated helped Valery consider that she 

could influence participation through the activities she planned. 

Valery continued to reconsider her role in supporting participation when she learned 

about the contrast between the students’ and her perspectives on pacing participation. The 

examples in Table 4.6 illustrate these perspectives. 

 

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Table 4.6 Example of pacing participation 

Student perspectives 

Teacher perspectives 

Harper:  Yo quiero hacer otra cosa cuando ya sé lo qué hacer… 

Valery: Les pido que me muestren con sus pulgares si 

si yo sé qué debo hacer en el libro o yo sé qué hacer 

entendieron, o más o menos, o no mucho. Y ahí, si 

cuando tenemos como un problema, si Señora Abad 

muchos están confundidos, explico de una manera 

explica más y más y más, yo ya quiero hacer la cosa. 

diferente o a lo mejor me devuelvo un poquito a algo 

 

I want to do something else when I already know what I 

que necesitamos repasar otra vez… [A veces] se 

have to do. Like, if I know what I have to do in the 

distraen o se cansan y preguntan ‘¿qué tengo que 

textbook or I know what to do when we have like a 

hacer? ¿cómo se hacía? ¿cómo era?’ Entonces si no 

problem, if Mrs. Abad keeps on and on and on 

están prestando atención, yo les pregunto a ellos: 

explaining, I just want to go for it. 

‘¿cómo vamos a saber qué hacer si no tengo todos los 

Emma:  Me gusta cuando hacemos lo difícil porque es más 

ojos aquí?’ y ahí ya puedo volver a explicar. 

divertido… pero a veces Señora Abad dice: ‘No pases 

 

I ask them to show me with their thumb if they 

la página. No haces esa página hoy. Haces ésta.’ Y a yo 

understood well, or more or less, or not so much. And 

no me gusta porque sólo necesitas repetir y repetir y 

then, if too many of them are confused, I explain in a 

repetir algo que ya sabes. No necesitas pensar. 

different way or maybe I go back a little bit to  

 

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Table 4.6 (cont’d) 

I like it when we do difficult things because it’s more 

something we need to review again… [Sometimes] they 

fun… but sometimes Mrs. Abad says ‘Don’t turn the 

get distracted or tired and ask ‘what do I have to do? 

page. Don’t do that page today. Do this one.’ And I 

How do you do this? How was it?’ So, if they are not 

don’t like that because you just need to repeat and 

paying attention, I ask them ‘how will we know what to 

repeat and repeat something that you already know. You 

do if I don’t have all eyes here?’ and right then and 

don’t have to think. 

there I can explain again. 

 

 

 

 

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Students concurred that excessive teacher explanations held them back and prevented 

them from engaging in more meaningful participation. For Harper, her sense of participation was 

compromised when she was expected to listen to Valery when she was ready to engage with the 

mathematics problem on her own. Emma’s comment suggests a desire to enliven participation by 

being allowed to move to more complex topics ahead in the book. For these students, 

participation promoted by students’ sense of autonomy and curiosity was often disfavored by the 

culture of participation based on repetitive and unappealing activities. 

While some students saw some repetitions as redundancies that undermined their 

participation, Valery saw those repetitions as a way to support participation. In contrast to 

students’ appreciation of autonomous exploration of mathematical ideas at their own pace, Valery 

seemed to discourage what she perceived as discordant involvement in mathematical tasks. For 

Valery, her role involved ensuring a uniform pace that allowed her to verify whether students 

were keeping up with specific mathematical ideas before proceeding to others. This meant 

bringing all students’ attention to the same explanations and activities to ensure everybody was 

ready to participate. 

As Valery’s awareness of these contrasting perspectives on pacing increased, she began to 

develop a more flexible consideration of the role she played in students’ participation. In addition 

to taking into account the ways of participating that specific activities could enable, Valery began 

to think of the content of those activities. Rather than understanding her role as pacing 

participation around predetermined ideas, she started to wonder how she could join students in 

understanding and developing the multiple ideas that could emerge in interactions. 

Valery’s openness to learn from students helped her broaden her perspectives on 

participation, approaching it with a willingness to expand what she saw as participation and what 

 63 

she considered influences on participation. Rather than having definitive visions of what 

participation looked like, Valery began to acknowledge that there were multiple interpretations of 

where participation was located, from participation identified in specific individual talk at 

specific moments, to participation through observation, thinking, gesturing and drawing that was 

part of social interactions at different paces during the span of a meaningful activity. The ways 

Valery began to talk about participation resonate with ideas from hermeneutic listening for 

participation as she highlighted how her students’ perspectives were influencing her own 

perspectives. Curiosities, questions, tensions and ideas about this classroom’s situated 

participation motivated and informed our collaboration to connect Valery’s reflections on 

participation with how she prepared to recognize and validate multiple unanticipated ways of 

participating, as I describe in Chapter 5. 

Reflections on My Evolving Perspectives on Participation 

Recognizing that my own connotations of participation could influence the culture of 

participation and how I made sense of the students’ and the teacher’s perspectives, I engaged in a 

reflection to bring to my awareness my own perspectives on participation. 

Initially, I valued listening and observation, which might have made students’ listening 

and observation in Valery’s classroom salient to me. As a student, my preference was not to 

speak up in classes as much as possible but to add to discussions if and when I thought I had an 

idea or question worth sharing. Although I never felt that I was not participating or that my more 

talkative peers showed more or better participation, I was aware of the importance that peers and 

instructors tended to place on talk. As an instructor in mathematics teaching methods courses at 

the college level, I noticed differences in student talk and I explicitly brought this issue to my 

students. In different semesters, I administered a survey. Reflecting on the questions, I used 

 

64 

talking and participating interchangeably and I was looking at participation in terms of present or 

absent student talk. In Valery’s classroom, I might have reinforced the existing culture of 

participation that favored student talk when I facilitated number talks while trying to manage 

talking time among students. 

My researcher role influenced my perspectives on participation, as I had read about 

classroom participation and how others have studied this phenomenon. Like other researchers 

(see for example Civil & Planas, 2004; M. B. Wood & Hackett, 2018), however, a definition of 

participation as students talk failed to help me make sense of participation as it emerged in 

Valery’s classroom. Additionally, I had researched the interplay between multiple semiotic 

resources and language in Valery’s classroom in the first year of our collaboration. This may 

have contributed to the impact that students’ comments on non-spoken participation had on me. 

This brief but important self-reflection helped me acknowledge my own influence on the 

culture of participation in Valery’s classroom. I also became aware that my perspectives on 

participation seemed closer to Valery’s than to the students’, which added to the synergies 

between our curiosity about and commitment to exploring students’ perspectives. 

 

 

 

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Chapter 5 Preparing to Recognize Diverse Participation 

The process of learning to listen hermeneutically for situated participation began with 

Valery’s reflection on her and the students’ perspectives on participation described in the 

previous chapter. This process continued with the co-design of iterative cycles of interventions to 

connect Valery’s reflection to her practice. This iterative co-design relates to factors that 

supported the teacher to learn to listen hermeneutically for participation, such as the role that 

different types of lessons and mathematical activities played in how students participated. It also 

fostered the teacher’s flexible adjustment of tasks, lessons and units in response to our growing 

understanding of student participation. 

Consistent with PDR principles, each intervention cycle included a set of commitments 

and actions intended to transform the culture of participation toward being more inclusive. We 

approached each intervention as an evolving opportunity to join students in reinterpreting 

participation. The design of the interventions was flexible and each cycle informed the following 

one by responding to ongoing reflections and discussions with students. Valery used the first 

lesson of each unit to elicit students’ understandings of the topic and to observe initial ways of 

participating, which informed the development of the rest of the unit. 

The focus of each cycle responded to Valery’s interests and how she was making sense of 

the evolving culture of participation. For the first cycle, she prioritized addressing aspects that 

students thought stifled their participation. For the second cycle, Valery and I talked about how 

the culture of participation was still overemphasizing student talk, so the focus of this cycle was 

on non-spoken participation. For the third cycle, Valery was interested in addressing feelings that 

students associated with participation and she brought up the idea of the positive feelings that 

 

66 

students mentioned when they described helping others. Collaboration among students became 

the focus of the third cycle.  

For each cycle, I discuss motivations and commitments, resources we drew on, the 

content of the unit, and focal teaching strategies intended to address different aspects of the 

culture of participation. Some of these strategies emerged in interaction, as Valery recognized 

and amplified participation in the classroom. We named those strategies after seeing them 

emerge across several interactions, and Valery continued to draw on them throughout the unit. 

Other strategies were designed and named as Valery and I discussed ways to address questions 

and tensions that emerged during her initial reflection. These designed strategies evolved over 

time, as Valery adapted them in the classroom. Although I describe these strategies in each 

intervention cycle, we used them cumulatively. For example, the strategies from the first cycle 

were also part of the second and third cycles. I also discuss how the design of interventions 

became progressively more open-ended, as Valery’s evolving conceptualization of participation 

and connections to practice became increasingly responsive to moment-to-moment interactions. 

First Intervention Cycle: Addressing Aspects that Stifled Participation 

In this intervention cycle, the teacher wanted to address the two main issues that students 

reported as stifling participation: overdoing certain activities and topics, and the competitiveness 

dominating whole class discussions. The goals of the intervention were to provide opportunities 

for students to extend their mathematical thinking and to better support students to share ideas 

with the whole class. 

Ideas from practitioner-oriented articles and knowledge from Valery’s previous 

professional development informed the design. Regarding the goal of providing opportunities to 

extend mathematical thinking, we initially talked about planning extension activities (Gearhart & 

 

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Saxe, 2014). Valery also shared ideas from a professional development workshop about allowing 

time for students to make sense of problems on their own without the teacher explaining or 

directing. Regarding the goal of supporting students to share ideas with the class, when preparing 

the research collaboration, Valery and I had discussed ideas about orchestrating mathematical 

discussions (M. S. Smith, Hughes, Engle, & Stein, 2009). Some of these ideas, however, focused 

on student talk sequenced by the teacher, which conflicted with our focus on multiple ways of 

sharing ideas that may emerge in unexpected places and times. We discussed modifications and 

alternatives to whole class discussions, including using writing to engage students with others’ 

ideas (Butman, 2014), and strategies to “narrow participation gaps” in whole class discussions 

(Hand, Kirtley, & Matassa, 2015). 

To make sense of how ideas from these articles could help us address Valery’s concern 

with the aspects that students reported as stifling participation, we analyzed the teaching unit as 

described in the textbook and the pacing guides. The pacing guides allotted two weeks for the ten 

lessons in this unit on multiplication. This was the second unit on multiplication in the academic 

year. Understanding the expected learning goals became important for the design of this and 

subsequent intervention cycles, as Valery began to think of including learning goals not included 

in the pacing guides as part of the extension activities. The first multiplication unit took place 

during the fall and it included three main objectives: (1) understanding meanings of 

multiplication, specifically repeated addition and counting objects in a number of groups of equal 

size; (2) developing proficiency multiplying two single-digit numbers and multiplying a single-

digit number and a multiple of 10; and (3) understanding the commutative, associative and 

distributive properties of multiplication. This second multiplication unit included lessons that 

addressed these three goals and also lessons on one and two-step word problems. 

 

68 

For the first lesson of this unit, we planned on identifying ideas that could help us address 

students’ comments about exploring what they called “interesting topics.” The class revisited 

meanings of multiplication in a whole class discussion that the teacher initiated asking “¿Qué es 

multiplicación?” [What is multiplication?]. The class continued with a number talk intended to 

elicit multiplication strategies. We anticipated that students would use direct modeling and 

number facts to explain how they solved the problems. We considered prompting the use of 

derived facts to extend students’ thinking. After this lesson, Valery and I decided to integrate the 

following two strategies, inspired in hermeneutic listening, in the rest of the unit. 

Focal strategy I: Addressing wonderment questions. During the first lesson of the unit 

a student asked about multiplying two two-digit numbers which started an impromptu whole 

class discussion (presented in the next chapter). After the lesson, Valery and I discussed that 

asking questions was one salient way of participating that students brought up in the focus 

groups. I shared with Valery the idea of wonderment questions. Instead of asking factual 

information or confirmation on the correctness of an answer, wonderment questions indicate 

curiosity about an unexplored topic (Aguiar, Mortimer, & Scott, 2010; Chin & Brown, 2010). 

This includes concepts not yet understood, and ‘what if’ questions about changing the conditions 

or context of a concept. 

We modified our initial idea of developing extension tasks and instead decided to address 

wonderment questions related to the topic of the unit as they emerged. Following the student’s 

question about multiplying two two-digit numbers, addressing wonderment questions meant 

reframing the learning objectives of the unit to include multiplying two two-digit numbers. We 

modified the initial unit outline, starting with a lesson inspired in a number talk where students 

are presented a 8x25 grid (Parrish, 2010) and answered the question “¿Cómo podemos saber 

 

69 

cuántos cuadritos hay?” [How can we know how many squares are there?]. To lessen students’ 

perception of over-competitiveness during whole class discussions, Valery and I designed the 

following teaching strategy. 

Focal strategy II: Gallery walks. As an alternative to whole class discussions, Valery 

used gallery walks where she displayed student work around the classroom, usually in posters 

that grouped similar solutions or strategies. Students walked around the classroom, observing the 

posters. There were sticky notes and pencils by the posters so that students could write their 

reactions, including asking questions, expressing agreement or disagreement, noting connections 

between strategies in different posters, or explaining the work displayed. Finally, students went 

back to their own posters and read others’ reactions. We tried different approaches to authorship, 

from asking all students to write their names on the papers to be displayed, to telling students 

they could decide whether or not to write their names, and whether or not to have their work 

displayed. 

Gallery walks were intended to mitigate the over-competitiveness that students reported 

experiencing in whole class discussions. Instead of regulating the order of sharing and limiting 

the number of students who got to share their work, in gallery walks all students had 

opportunities to consider everybody else’s mathematical ideas. Sometimes there was a brief 

whole class discussion following a gallery walk, where students shared what they noticed or 

what strategies they found interesting or surprising. Other times, a task followed where students 

were asked to try a strategy that they had seen in the gallery walk. 

Second Intervention Cycle: Recognizing Multiple Ways of Participating  

This intervention cycle focused on students’ non-spoken ways of participating, including 

gestures, drawings, writing, and movement. The goal of the intervention was to support the 

 

70 

teacher to learn to see students’ non-spoken participation as valid and contributing to the ongoing 

mathematical activity in the classroom. At this point, Valery was committed to recognizing 

multiple ways of participating besides student talk. Ideas about gesturing and using specific 

materials emerged in conversations with two students during the first intervention cycle. During 

a whole class discussion in the multiplication unit, one student, Daniel, was folding a piece of 

paper under his desk. Valery noticed it and we discussed that Daniel could give us ideas for a 

multiplication task using paper folding. In my following visit to the classroom, Daniel was 

folding paper under his desk again. When I asked him if I could see what he was doing, he 

quickly responded “nada” [nothing], as he put the paper in his desk and opened his mathematics 

textbook. It took some talking to convince Daniel that I was not trying to catch him out of task, 

that I was authentically curious about what he was doing, and that the teacher and I thought 

paper folding could be part of a mathematics lesson he could help us launch. When I described 

this interaction to Valery, she became increasingly interested in validating the exploration of 

mathematical concepts associated with non-spoken actions. 

Valery and a student, Coretta, had another conversation that further reinforced this 

interest in exploring non-spoken ways of participating. During a lesson, Coretta stretched her 

fingers and looked at them quietly before answering questions about word problems. When 

Valery approached her and asked if looking at her fingers helped her with the problems, Coretta 

said “Sí. Para ver la respuesta. A veces mis manos hacen cosas que yo no sé qué son” [Yes. To see 

the answer. Sometimes my hands do things that I don’t know what they are]. This comment added 

to Valery’s interest in exploring non-spoken ways of engaging with mathematical tasks. 

We drew on practitioner-oriented articles on non-spoken mathematical communication 

(Hand et al., 2015; J. N. Thomas, 2018; Wilson, 2012). We discussed gesturing, drawing and 

 

71 

movement as ways to generate and animate mathematical ideas, instead of incidental to or a 

replacement of spoken communication. Additionally, we discussed examples of classroom 

interactions when Valery had explicitly referred to students’ gestures and handling of objects. For 

example, I brought up a geometry lesson from the previous year, where Valery drew attention to 

a student’s swiping movement with an arm to describe the tracing of an angle. Valery was 

particularly interested not only in her own recognition of students’ non-spoken participation but 

also on students’ recognizing each other’s gestures and body movement as part of the exploration 

of mathematical ideas. 

The pacing guides allotted two weeks for the ten lessons in this unit on fractions. There 

were four main objectives: (1) understanding a fraction as a part-whole relationship and as a 

number in the number line; (2) representing fractions using set models, area models and fraction 

bars; (3) comparing fractions, including comparing whole numbers to fractions and equivalent 

fractions; and (4) solving word problems that involved representing and comparing fractions. 

The unit focused on proper fractions and representing whole numbers as fractions. During the 

first lesson of the unit, Valery showed the fraction 1/4 written in a poster sheet and asked the 

class “¿Qué sabes de fracciones?” [What do you know about fractions?]. She asked students to 

write or draw anything they wanted to share about fractions on sticky notes and to place their 

notes on the poster. Following this activity, she read out loud some of the ideas on the sticky 

notes related to definitions of fractions. She then asked “¿Qué son las fracciones?” [What are 

fractions?] in a whole class discussion. The class revisited this poster throughout the unit, 

sometimes adding or revising the responses on the sticky notes. 

We initially planned using materials such as paper and pencil, counters, geoboards, and 

unifix cubes to foster the use of gestures, drawings, and movement as ways of participating. In 

 

72 

the initial lessons of the unit, however, some students did not use paper to draw fractions, as 

Valery intended. Instead, they folded the paper to represent a fraction. In the second part of the 

unit, we designed lessons around a representation of Michigan demographics (see Appendix C). 

Both in the paper folding and in the demographics tasks, Valery drew on the following teaching 

strategy that related to hermeneutic listening’s idea of amplifying participation. 

Focal strategy: Spotlighting gestures, drawings, and movement. One way in which 

Valery spotlighted gestures, drawings and handling of objects involved showing these actions to 

the class to focus attention on specific concepts. For example, when students folded paper to 

represent and compare fractions, Valery drew attention to how a student was trying to be precise, 

making sure the paper was “doblado perfecto” [folded perfectly]. A follow up task emerged about 

the notion of the parts of a fraction being the same size. Another way Valery spotlighted non-

spoken participation involved showing students’ actions as part of a problem-solving strategy, 

which I describe in chapter 6. 

Third Intervention Cycle: Encouraging Collaboration Among Groups 

This intervention focused on helping as a legitimate way of participating. The purpose of 

the intervention was to validate and foster collaboration, which was informed by students’ focus 

group responses, reflections on previous intervention cycles, and a conversation with a student. 

The idea of helping that emerged during focus groups became a recurrent topic of discussion 

during reflections. When Valery and I revisited ideas from our co-analysis of focus group 

responses, Valery mentioned that the first two interventions had addressed the issue of multiple 

ways of participating and relating the individual and the social. She wondered whether and how 

we could still address feelings associated with participation and the qualification of participation. 

The idea of supportive and productive help, endorsed by students, became relevant. In this 

 

73 

discussion, I brought up listening and observing as ways of participating that students had 

mentioned and that we had not elicited or recognized. 

A conversation with one of the students during the second intervention resonated with 

these reflections. During the second intervention, Valery thought that Calum was not interested in 

one of the tasks because he left his group and stared out the window. Valery was particularly 

intrigued because this was unusual of Calum, and we decided that I would talk with him about 

his participation in that task. When I asked Calum about the moment when he left his table, he 

explained he was not bored. He thought the task was difficult, which he liked. He explained he 

was not just staring out the window: he was “más o menos como haciendo trampa” [kind of 

cheating]. His group was stuck and he thought he’d take a look at how other groups were 

approaching a problem. He thought that if he had looked at a group’s work, that would have been 

considered cheating. He decided to stand close to a group to eavesdrop, looking out the window 

to avoid appearing as if he were cheating. When I shared what Calum said to me with Valery, we 

discussed ways to transform perceptions of what Calum did from cheating to collaborating. 

We drew on practitioner-oriented articles about geometry teaching that resonated with 

ideas of multiple ways of participating (Mack, 2007; Tent, 2001). We also discussed ideas about 

collaboration in mathematics classrooms (Ghousseini, Lord, & Cardon, 2017) and we revisited 

the idea that one of the students, Juan, had mentioned during the focus groups about not only 

helping but also asking for help as an important way of participating. There were 12 lessons in 

this unit and the main objectives were: (1) categorizing shapes according to relevant attributes 

(e.g. number of sides) and characteristics (e.g. absence or presence of parallel sides); (2) solving 

problems about the area and perimeter of rectangles; and (3) classifying and comparing angles. 

For the first lesson of the unit we adapted Mack’s (2007) task to elicit students’ understanding of 

 

74 

two-dimensional shapes and foster collaboration among students. Each student received a card 

with a shape in it. Students walked around the classroom finding others whose shape belonged in 

the same category. A whole class discussion ensued, where Valery elicited students’ reasons for 

their categorizing and naming of the shapes. The unit continued with lessons about the names 

and characteristics of different polygons (triangles, rectangles, pentagons, hexagon, heptagon, 

octagon). In the initial lessons, the teacher and I planned tasks where students would collaborate 

within small groups to make, name and describe shapes. 

As the content of the unit deviated from the pacing guides and became progressively 

more unpredictable than in previous cycles, Valery and I relied on each other’s expertise and 

ideas to refocus the lessons. Motivated by students’ wonderment questions, in the second part of 

the unit the class explored properties of shapes. One lesson focused on the triangles of a polygon, 

with students working in groups exploring the relationship between the number of triangles that 

made up a polygon and its number of sides. The final part of the unit, that originally focused on 

defining and exploring strategies to calculate the area and the perimeter of rectangles, evolved to 

focus on exploring circles (as discussed in the next chapter). Throughout the unit, Valery 

continued to refine how she was integrating the idea of collaboration, observation and listening 

through the following focal strategy, related to hermeneutic listening’s idea of engaging in 

participatory interactions to influence lessons. 

Focal strategy: Group ambassadors. Valery proposed the idea of group ambassadors, as 

a teaching strategy to foster collaboration both within and among groups. Students started 

working in groups and once Valery considered groups had come up with strategies that they 

wanted to try, she asked each group to choose an ambassador. The ambassador went to another 

group and observed, while groups resumed their work. After an observation period, sometimes 

 

75 

Valery left it to each ambassador to decide if or when to intervene, ask questions or contribute 

ideas to the host group. Other times, she asked ambassadors to share how their original group 

was approaching the problem. Ambassadors then reported back to their original groups. Each 

group decided whether to adjust their approach to the problem taking into account the ideas the 

ambassador reported. With this strategy, Valery acknowledged observation and listening in as 

valid and legitimate ways of participating, while simultaneously fostering opportunities to help 

and ask for help without stifling participation. 

 

 

 

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Chapter 6 Connecting Reflection and Practice 

During the implementation and ongoing analysis of interventions, the connections 

between reflection and practice grew stronger and more evident, creating a synergy benefiting 

both processes. The intervention designs, described in the previous chapter, reflected Valery’s 

vision for transforming the classroom culture of participation. Rather than understanding her role 

as following prescribed steps intended to facilitate participation, she responsively drew on her 

evolving repertoire of teaching strategies. Students responded by amplifying each other’s ways 

of participating, joining Valery in legitimizing them. Creative ways of exploring unanticipated 

mathematical ideas emerged, intensifying Valery’s openness when interpreting participation in 

times and spaces that she had not considered before. She continued to add complexity to her 

reconceptualization of participation, simultaneously recognizing the role of meaningful and 

flexible activity, multiple semiotic resources, and the influence of school demands. 

In the following sections, I describe three transformations that the teacher, the students, 

and I experienced as we entered the implementation phase of the mathematics units: (1) 

expanding the scope of the mathematical ideas available for exploration, (2) using multiple 

semiotic resources to develop mathematical ideas, and (3) participating fluidly in mathematical 

activity. I draw on classroom examples from the video-recorded lessons that Valery and I co-

analyzed, as well as examples from other lessons that helped understand each transformation. I 

discuss temporal and spatial dimensions of the kind of participation Valery was learning to see as 

well as factors influencing her hermeneutic listening for participation, including Valery’s 

evolving role, the types of lessons and mathematical activities, and the flexible adjustment of 

tasks, lessons and units in response to student participation. After discussing interpretations of 

each example, I describe transformations from a semiotic innovation perspective. I describe 

 

77 

Valery’s ongoing reflection on participation and her identification of tensions among her 

evolving reconceptualization of participation, and her evolving practice and demands from the 

school context. To do this, I draw on quotes from the co-analysis sessions. 

Expanding the Scope of Mathematical Ideas Available for Exploration 

Over time, Valery joined students in exploring mathematical ideas they were interested 

in. An interaction from Fall 2016, where the class was developing different representations of 

1/10, illustrates Valery’s prior approach to exploring mathematical ideas: 

Table 6.1 Valery's previous approach to topics to explore 

 

Utterances 

Translation 

Actions 

Billy 

¿Puede ser al revés, como 

Can it be the other way 

With arms 

 

 

 

 

 

 

no uno y diez pero diez y 

around, like, not one and 

stretched out 

uno? Como, ¿una fracción 

ten but ten and one? Like, 

toward the board 

puede algo así como tiene el 

can a fraction like have a 

where 1/10 was 

numerador, cómo se dice 

numerator, how do you say 

written, moves 

greater than, el 

denominador? 

greater than, the 

hands up and 

denominator? 

down. 

Valery 

Ah! Eso lo vas a ver no el 

Oh! You’ll study that not 

 

 

próximo sino el siguiente 

next year but the year after 

año con Señora M. 

with Mrs. M. 

 

 

 

Claire 

Estamos en fracciones 

We’re studying unit 

Looking at Billy. 

 

unitarias, Billy. 

fractions, Billy. 

 

Valery 

Por ahora estamos viendo 

Right now we’re studying 

Nodding. 

fracciones unitarias. 

unit fractions. 

 

78 

Valery’s response to Billy and Claire’s reminding him of the topic to be studied indicate 

that the class as a whole had established a border separating the mathematical ideas they could 

explore at a given time from the ones that were off-limits. During the intervention cycles, Valery 

blurred this border by expanding the scope of ideas the class considered as possible and relevant 

for their learning. Instead of pacing guides mandating mathematical content and Valery 

assuming the role of an enforcer of this content, alternative unexpected topics emerged in the 

class. The perspectives from the student focus groups regarding feeling ready and curious about 

topics beyond the scope of a lesson sensitized Valery to notice and explore unexpected ideas 

with her students. As Valery developed lessons from these ideas, students became increasingly 

explicit in their suggestion of ideas. Although Valery and I initially co-planned lessons that 

addressed topics predetermined in the pacing guides and the textbook, topics evolved in response 

to students’ declared interests. 

In the following sections, I describe how addressing students’ curiosities and recognizing 

mathematical ideas in off-task comments motivated participation in mathematical activity that 

involved ideas meaningful to both students and Valery. 

Addressing students’ curiosities. In addition to using questions to elicit students’ 

understanding, Valery engaged in interpreting the wonderings that students’ questions 

communicated. In turn, students recognized that certain questions could influence the ideas the 

class explored. Considering students’ wonderment questions became a legitimate way of 

participating that steered the direction of a unit beyond the scope of the content predetermined in 

the pacing guides. This process started in the first lesson of the first intervention, when the 

following interaction emerged during a whole class discussion on what it means to multiply: 

 

 

79 

Table 6.2 Example of wonderment question 

 

Utterances 

Translation 

Actions 

Calum 

Tengo pregunta. ¿Qué es cien 

I have a question. What’s one 

Looking at Valery. 

por cien? ¿Es diez mil? 

hundred times one hundred? Is 

 

 

 

Jimmy 

Mil. 

it ten thousand? 

One thousand. 

 

 

 

 

 

 

 

 

 

 

 

Valery 

¿Escuchas la pregunta de 

Can you hear Calum’s 

Addressing the 

 

Calum? 

question? 

class. 

Calum 

¿Qué es cien más cien? Es 

What’s one hundred plus one 

Speaking up, 

 

diez mil. 

Jimmy 

Mil. 

hundred? It’s ten thousand. 

addresses the class. 

One thousand. 

Lia 

No, no es mil. 

No, it’s not one thousand. 

Calum 

Yo pienso es diez mil. 

I think it’s ten thousand. 

Valery 

¿Por qué? 

Why? 

Calum 

Porque no puede ser mil 

Because it can’t be one 

 

 

 

porque mil es cien por diez. 

thousand because one 

 

 

thousand is one hundred times 

ten. 

Halley 

Tú sabes la respuesta. 

You know the answer. 

Points at Valery. 

 

Y tú. Dinos. 

And you. Tell us. 

Points at José. 

Valery 

Pero, ¿cómo sabes la 

But how do you know the 

respuesta? Vamos a tener que 

answer? We’ll have to do 

hacer más de esto después. 

more of this later. 

 

80 

Unlike her earlier concern for isolating content by grade level, Valery engaged the class 

with Calum’s wonderment question. As more students showed curiosity and shared ideas, Valery 

became increasingly interested in this topic. After the lesson, Valery and I discussed how 

Calum’s question could inform future lessons. She decided to introduce multiplying a two-digit 

number by a single-digit number, which was beyond the scope of the pacing guides. The class 

engaged in a task figuring out the number of cells in a 25x8 grid. Using a gallery walk—one of 

the strategies we co-designed as an alternative to whole class discussions— students compared 

different strategies. They assessed the efficiency of counting by ones, repeated addition, and 

splitting the problem into smaller parts. Eventually, the class converged to the idea of using facts 

they knew to split the problem into smaller parts (e.g. 4 x 25 equals 100 and 2 x 100 equals 200). 

Toward the end of the unit, Valery revisited Calum’s question, asking the class to figure 

out ways to know what one hundred times one hundred is. During this whole class discussion, 

Calum was quieter than usual. As students where putting away materials immediately after the 

lesson, the following interaction took place: 

Table 6.3 Calum's sense of participation 

 

Utterances 

Translation 

Halley 

Esto era muy difícil. A mí me gusta 

This was very hard. I like that a lot. That’s 

 

 

 

mucho. Esto es por qué yo mostré tantas 

why I showed so many strategies. What 

estrategias. ¿Qué de ti, Calum? Hoy no 

about you, Calum? You didn’t participate 

participaste. 

today. 

Calum 

Todo que hicimos hoy fue para responder 

Everything we did today was to answer 

mi pregunta ‘qué es cien por cien’ que yo 

my question ‘what is one hundred times 

quería saber. 

one hundred’ that I wanted to know. 

 

81 

Drawing on the idea from focus groups of considering qualities instead of quantity of 

participation, Valerie and I reflected on Calum’s statement in terms of the relative importance of 

the number and length of his speaking turns vis-à-vis the influential reach of his question by 

reshaping the topic of the entire lesson. For him, influencing the topic the class was studying 

constituted a personally important way of participating. This way of participating highlights the 

temporal dimension of participation extending beyond the utterance or lesson level. Calum’s 

question had enduring effects for the class and for him, as he experienced a sense of mutual 

influence even in this last lesson of the unit when he was quiet. This co-interpretation of this 

brief interaction encouraged Valery to continue exploring wonderment questions as a way of 

challenging her interpretations of participation on the basis of quantity. 

Although I focused on Calum’s example, there were other instances of wonderment 

questions that the class took up. In one lesson where the class was working on a task about 

recognizing fractions as numbers on the number line, Lia asked whether fractions could be 

added, since they were numbers. The topic of adding fractions became part of the unit, extending 

topics from the pacing guides that considered representing and comparing fractions only. In 

another lesson, Rose asked what were “esos números antes de cero” [those numbers before zero], 

referring to the negative numbers on a number line displayed on the wall above the board. 

Negative numbers became the exploration of a series of number talks. 

Students picked up on Valery’s interest in addressing their curiosities and Valery 

perceived an increase in the number of wonderment questions asked. Valery and I wondered 

whether this perceived increase in student questions was related to students noticing that Valery 

was now taking up and exploring their questions, or if the number of wonderment questions was 

stable but Valery recognized them more often now than before. My analysis of the entire video 

 

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data set suggests that students gradually became more insistent when asking these types of 

questions. Two examples include Juan asking if they could study the abacus and Coretta asking 

how it was possible for her older sister to add and subtract letters and numbers. Valery briefly 

entertained their questions, asking students what they knew about the abacus and asking their 

conjecture about what letters represented in mathematical operations. She announced she would 

take note of the questions and would try to address them when they had time. A few weeks later, 

Juan insisted on his question suggesting he could bring an abacus to school for an activity. 

Stacey and Marta brought up Coretta’s question, asking if they could talk about her question 

“como hablamos sobre que Calum quería” [the way we talked about what Calum wanted]. In 

both cases, students drew attention to wonderment questions as possible sources of topics to 

explore. In Coretta’s case, other students showed interest in her question and directly asked for 

the question to be given the same importance that Calum’s question received. Calum’s question 

and how Valery addressed the question continued to influence students’ participation over time 

beyond that specific interaction. 

Semiotic innovation that facilitated addressing students’ curiosities involved the 

interaction among challenging the authority of tradition, shifting roles and reassigning saliency 

to questions. Sticking to the goals predetermined in the pacing guides, even if they did not reflect 

the students’ interests, was a strong tradition that Valery used to observe. Valery dared to 

challenge this tradition by expanding the learning goals. This involved Valery shifting away 

from her role of safeguarding predetermined content and, instead, looking for connections 

between predetermined content and the ideas students were interested in exploring. Students’ 

roles also shifted, as they started to become co-participants in determining the content of lessons. 

 

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Valery reassigned saliency to wonderment questions as a legitimate way of participating and 

students responded favorably by engaging in this way of participating. 

When we discussed the process of addressing wonderment questions, Valery expressed a 

tension related to school expectations, specifically assessment. Valery could decide some of the 

assessment for each unit, but other parts were predetermined by the school. Although she was 

curious about and thought the class benefitted from addressing wonderment questions, she 

frequently felt the need to filter those questions in light of the school assessments. Valery felt 

that she should address only student ideas that would help them during the school assessments, 

even if unanticipated explorations were more sophisticated than the predetermined content. As 

she joked: “no va a importar que saben hacer un edificio entero, si la prueba sólo quiere ver que 

sepan abrir la puerta del edificio” [it won’t matter if they know how to build an entire building if 

the test only cares about whether they can open the building’s door]. Valery looked for ways to 

connect the explorations of students’ wonderment questions to opportunities for them to better 

understand topics that would be assessed. In the multiplication unit, for example, she explicitly 

expressed that multiplying a two-digit number by a one-digit number could involve multiplying 

two one-digit numbers, which was the topic in the assessment. In other cases, however, she felt 

the class would need to use their time to address wonderment questions besides studying 

predetermined topics. 

Recognizing mathematical ideas in “off-task” behavior. Throughout the interventions, 

Valery recognized mathematical ideas in seemingly off-task behavior. These ideas pushed the 

boundaries of the mathematical content that the class explored. Over time, students began to 

bring attention to mathematical ideas in jokes, side conversations and games. I illustrate this 

transformation with an example from the third intervention cycle. The planed content of the 

 

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geometry unit focused on defining and calculating the area and perimeter of rectangles. In this 

lesson, students had solved examples of the area and perimeter of rectangles on the board. As 

Valery turned to her desk to get materials to transition to a new task, Willie went to the board 

and grabbed a marker: 

Table 6.4 Willie's strategy for the area of circles 

 

Utterances 

Translation 

Actions 

Willie 

Y el círculo es como un 

And the circle is like a zero 

Smiling as he draws a 

 

 

 

cero porque no tiene área y 

because it doesn’t have 

circle on the board and 

perímetro. 

 

area and perimeter. 

writes a number zero in the 

 

middle of the circle. 

Valery 

Amigos, ¿vieron lo que 

Friends, did you see what 

Quickly turns around to 

 

 

 

 

Class 

 

hizo Willie? 

Willie just did? 

see what Willie drew. 

 

 

 

 

 

 

 

 

 

 

Facial expression of 

surprise, as she addresses 

the class. 

Some students giggle, 

others say no. 

Valery 

Mira lo que hizo aquí. 

Look what he did here. 

Pointing at the circle. 

 

No todos te oímos. 

Not everybody heard you. 

Looking at Willie. 

Willie 

¿Esto? 

 

 

 

 

Valery 

Aquí. 

This? 

 

 

Here. 

Stops smiling. Points at the 

rectangle problem he 

explained earlier. 

Pointing at the circle. 

 

85 

Table 6.4 (cont’d) 

 

Lo que acabas de decir. 

What you just said. 

 

Willie 

¿El círculo es como un 

The circle is like a zero 

Looking at Valery. 

 

 

Class 

 

 

cero porque no tiene 

because it doesn’t have 

perímetro y área? 

perimeter and area? 

 

 

 

 

 

 

 

 

A student gasps, others 

giggle, others exclaim 

‘what?!’ 

Valery 

¡Qué interesante! 

How interesting! 

Addressing the class. 

 

Valery and I interpreted this episode in three parts: in the first part, Willie seemed to be 

joking, smiling as he said something he found funny. In the second part, when Valery draws 

attention to the circle and to Willie’s comment, he seemed confused and tried to redirect the 

teacher’s attention to one of the problems he had explained earlier. In the third part, some 

students started considering what Willie said and expressed surprise. What started as a joke 

during the few seconds when Valery turned to her desk, evolved into a mathematical idea that 

Valery spotlighted and that generated surprised and interest. 

This interaction shifted the focus of the unit to the topic of circles. Willie’s comment 

motivated a series of lessons that started with a heated debate about whether the circle was a 

shape. Some students argued the circle was not a shape because a shape was enclosed by sides 

and circles did not have sides. These students argued that sides had to be straight lines. A new 

debate arose, with other students saying sides could be curved, to accommodate the case of the 

circle. Eventually, these debates cycled back to redefining perimeter as “la distancia alrededor de 

 

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una figura plana” [the distance around a flat shape]. Students thought this definition worked for 

both quadrilaterals and circles, as opposed to the textbook definition of perimeter as the sum of 

the lengths of a polygon’s sides, that they thought did not work for circles. As these debates and 

redefinitions evolved, the class engaged in figuring out the area and the perimeter of circles using 

their own definitions. 

When co-analyzing this episode and its influence on the direction the unit took, Valery 

reiterated her interpretation of specific interactions as sources of meaningful mathematical 

activity that enabled participation. She mentioned: “Ellos mismos dijeron que les gusta participar 

en cosas interesantes. Entonces hay que encontrar las cosas interesantes” [they themselves said 

they like to participate in interesting things. So we have to find the interesting things]. In the 

example, the ideas about circles, sides, area and perimeter that the class explored were topics that 

students considered interesting and that generated debates they did not want to stop. Referring to 

the initial outline of the unit, Valery mentioned: “Lo que seguía era más lecciones de área y de 

perímetro y ¿cómo iban a participar ahí? A lo mejor bien aburridos y sin creatividad porque no 

era nada nuevo” [What was coming up was more area and perimeter and, how were they going 

to participate in something like that? Maybe all bored and with no creativity because it wasn’t 

anything new]. In this quote, she acknowledges the relationship between the content of activities 

and how students participated. In contrast with her initial perspective of participation, instead of 

framing participation in terms of the absence-presence binary, she qualified participation, 

predicting they would have participated in apathetic and uncreative ways. Flexibly adjusting 

tasks, lesson and units in response to student participation emerged as a characteristic of the 

evolving culture of participation that fostered the sense of mutual influence that students valued. 

 

87 

There were other instances of mathematical topics that emerged from what could initially 

be perceived as off-task behavior. That was the case when Valery noticed that Daniel was 

folding paper during a whole class discussion and Valery and I talked about approaching him to 

learn about what he was doing. The following time I visited the classroom, Daniel was folding 

paper again. When I asked him to share what he was doing, Daniel hid the paper quickly 

thinking I was trying to catch him off-task. Eventually he shared what he was doing and Valery 

became interested in using paper folding for a task on multiplication. Daniel helped introduce 

this task to the class, as he showed his paper folding and Valery asked students to tell the number 

of parts there would be after each folding and to explain their conjectures. Another example 

comes from the fractions unit when a lesson using coins as fractions of a dollar emerged as Gina 

overheard two other students playing as if they were buying items from each other. In their 

game, Ada was buying from Mackenzie and she needed one dollar in change. As they both 

looked at the play money, Ada asked Mackenzie to give her “two quarters and five dimes” in 

change. Gina brought this to Valery’s attention, saying: “hay fracciones en monedas” [there are 

fractions in coins]. Valery eventually used play money in a lesson on fractions addition. Valery 

looked for and recognized participation in Daniel’s paper folding and in Ada and Mackenzie’s 

play, instead of isolating participation as an occurrence located in the space of a class discussion 

or small group work when students talk about mathematical ideas. 

Similar to addressing students’ curiosities, semiotic innovation also influenced the 

recognition of mathematical ideas in off-task behavior by reassigning saliency to students’ ideas, 

promoting role shifting, and challenging the authority of tradition. The exploration of circles 

emerged as Valery assigned saliency to a comment that Willie intended as a joke and not as a 

mathematical idea worth exploring. Instead of adopting the classroom management role of 

 

88 

enforcing classroom norms and minimizing off-task comments, Valery’s role shifted toward 

interpreting what Willie did as a way of participating that involved meaningful mathematical 

ideas. Willie’s role shifted too, from a joker or a disruptive student to authoring a mathematical 

claim that the class debated in several lessons. Valery noted that students at this school began to 

explore the area and perimeter of circles in fifth grade. Instead of delaying this exploration, 

Valery challenged the authority of tradition established by the pacing guides by joining her 

students’ interest in exploring circles. As she expanded the scope of the unit in this direction, she 

thought that exploring ideas about circles helped the class develop nuanced understandings of the 

concepts of sides, area and perimeter in polygons. 

This transformation presented a tension for Valery. In several instances, such as Willie’s 

example, Valery felt she put students on the spot to showcase their ideas. Valery wondered if this 

made some students uncomfortable, especially when students thought she was drawing attention 

to them because they were doing something unrelated to class. Valery related this concern to the 

social aspects associated to participation that students expressed during focus groups. She 

mentioned a tension between spotlighting mathematical ideas in off-task behavior and avoiding 

students’ embarrassment. 

Using Multiple Semiotic Resources to Develop Mathematical Ideas 

In the previous examples, Valery drew on two forms of student talk, wonderment 

questions and “off-task” comments, to expand the mathematical content the class considered. In 

addition to student talk, she began to recognize and draw attention to gestures, drawings and 

movement that she considered as relevant to the development of ideas. Both Valery and the 

students started to value these semiotic resources in mathematical activity. Recognizing multiple 

semiotic resources meant relocating participation from seeing participation in students’ 

 

89 

utterances to seeing participation in students’ gestures, movements, gaze, drawings, and handling 

of objects. 

One example of the recognition and uptake of multiple semiotic resources comes from 

the unit on fractions, in the second cycle of interventions. Valery handed out copies of graph bars 

representing different Michigan demographics (see Appendix C). Students worked in small 

groups, figuring out what fraction of the population corresponded to each category in the graphs 

(for example, which fraction of the population spoke a language other than English at home and 

which fraction spoke English). Because of the quality of the printing, the fill-in color of each bar 

appeared not as a solid color but as if it had been printed out in smaller sections, likely due to an 

ink cartridge that needed replacement. Valery and I did not notice this coloring issue but the 

students did, and many groups interpreted the broken coloring as showing each section 

partitioned into equal parts. To figure out the fractions, students counted the number of these 

parts in each bar. For the first graph, representing women and men, students counted more than 

50 equal parts. Valery tried unsuccessfully to elicit alternative strategies. In the following 

example, Ada and Willie were leaning over observing as Jimmy counted out loud the parts in the 

women’s bar and Marta counted the parts in the men’s bar. 

Table 6.5 Students' initial strategies to compare fractions 

 

Utterances 

Translation 

Actions 

Ada 

Señora Abad, ¿hay más 

Mrs. Abad, are there more 

Addressing Valery. 

 

mujeres o hombres? 

women or men? 

Valery 

Mmm. 

Mmm. 

Willie 

¿Quieres contarlos? 

Do you want to count them? 

 

 

 

Valery 

No, 

No, 

Shakes head, smiling. 

 

90 

Table 6.5 (cont’d) 

 

 

 

 

y por eso tengo una 

and that’s why I have a 

 

pregunta, Marta y Jimmy. 

question, Marta and Jimmy. 

Marta and Jimmy stop 

¿Cómo van a saber las 

How will you know the 

counting. 

fracciones? 

fractions? 

Ada 

Contando los partes. 

By counting the parts. 

Valery 

¿Qué mas podemos hacer 

What else can we do 

 

además de contar? 

besides counting? 

Jimmy 

Yo no sé, porque está difícil 

I don’t know, because 

 

contarlos. 

counting them is difficult. 

Marta 

Contar despacio y contar 

Counting slowly and 

 

 

 

 

 

 

 

dos veces. 

counting twice. 

Goes back to counting. 

 

In this example, Valery attended to what students were doing to make sense of how they 

were approaching the problem. She drew on spoken language to try to redirect the students’ 

approach. Students referred back to what they saw as parts in the graphs and continued their 

approach. Other questions Valery tried with other groups included asking for a strategy that 

would be more efficient than counting and asking if they could compare the two bars without 

counting the parts. In both groups students insisted in counting parts. 

Valery was having one of these conversations with a group when she saw Emi use her 

index fingers to compare the bars. Emi put the tip of one finger where the women’s bar started 

and used her other finger to indicate where the bar ended (as represented in Figure 6.1). Keeping 

this gesture (left finger stretched and right finger indicating where the bar ended), she slid her 

 

91 

hands to the bar representing men. She then took a pencil and wrote 1/2 above the women’s bar 

and 1/2 above the men’s bar. On a loud and excited voice, Valery called on the class and asked 

all students to congregate around Emi’s table. Valery handed a new sheet with the demographics 

graphs to Emi and asked her to show her strategy to the class: “haz de nuevo lo que acabas de 

hacer” [do again what you just did]. Quietly, Emi repeated her hand gestures and movements and 

looked up to Valery, without writing down the fractions this time. Valery remarked: “¡Qué 

interesante!” [How interesting!]. Students responded with unintelligible chatting as they went 

back to their groups and adapted Emi’s strategy. One group marked the length of the men’s bar 

on a piece of paper and slid the piece of paper to compare it to the length of the women’s bar. 

Another group used a ruler to measure each bar. Another group folded the graph in the middle 

where the women’s and the men’s bars met. They held the paper up, backlighting the graph to 

compare the two bars. 

Figure 6.1 Emi's gestures on the graph 

 

Emi contributed a generative strategy that influenced how others approached this 

problem. While Valery relied on spoken language in the interaction with Ada, Marta, Jimmy and 

Willie, Emi’s strategy relied on the interaction between her gestures and the graph. Instead of 

asking Emi to put in words what she did and how it helped her figure out the fractions, Valery 

 

92 

spotlighted Emi’s gestures on the graph. Students uptake of Emi’s strategy, adapting it in 

multiple ways, also involved interactions among students, the graph, and resources such as 

paper, rulers, and movement. Later in the lesson, students used similar strategies to compare bars 

in the other demographics graphs. 

When Valery and I discussed the interactions around Emi’s strategy, Valery established 

connections to her reconceptualization of participation. She recognized that Emi made sense of 

the problem through her quiet participation, which highlighted the idea of multiple ways of 

participating besides talk. As Valery put it, “no tuve necesidad de preguntarle [a Emi]” [I didn’t 

have to ask her]. This was particularly meaningful to Valery, as Emi tended to be quiet in whole 

class discussions. Valery and I also discussed ideas from the focus groups about the interplay 

between the social and the individual. Valery recognized that she used Emi’s gestures and 

writing to gain insights into Emi’s sense-making, but also to help Emi influence how the rest of 

the class approached this problem. She stated that the strategy stopped being just Emi’s and 

became part of the whole class: “ni tuve que preguntarles si lo que Emi hizo les iba a servir para 

la actividad. Ellos solitos se fueron a intentar” [I didn’t even have to ask them if what Emi did 

would help them to do the activity. They, pretty much on their own, went on to try]. She added: 

No era solamente como ver si Emi estaba participando o si estaba callada. Tenía que ver 

con que yo me diera cuenta cómo estaba participando, y que los niños se dieran cuenta y 

que hicieran algo con la idea de Emi. 

It wasn’t just like seeing whether Emi was participating or if she was quiet. It had to do 

with me noticing how she was participating, and with the children noticing as well, and 

that they did something with Emi’s idea. 

 

93 

In this statement, Valery connected ideas of multiple ways of participating, such as Emi’s 

gestures and writing, with qualifying participation (i.e. “how she was participating”), and with 

the interplay between the individual and the social. 

Beyond Emi’s example, recognizing and taking up multiple semiotic resources became 

part of the culture of participation. For example, in a lesson when the class discussed if 5/4 was a 

possible fraction, Valery saw Emma’s gestures. Emma traced in the air an arc that would 

correspond to a quarter of a circle, stopping momentarily before tracing the second arc in the air, 

then the third and the fourth. She paused a little bit longer this time and then traced a fifth arc in 

the air, as if retracing the first one she gestured. Valery interpreted this to mean Emma thought of 

four fourths making up a whole circle and then an additional fourth. Valery mimicked Emma’s 

gestures to bring Emma’s idea to the attention of the class. A student, Lia, connected these 

gestures to the idea of time: “cinco cuartos de hora es algo que existe. Es como setenta y cinco 

minutos. Entonces cinco cuartos es algo que existe.” [five quarters of an hour is something that 

exists. It’s like seventy-five minutes. So, five quarters is something that exists]. Emma’s gestures 

helped Lia develop an idea about whether the fraction 5/4 was possible or not. Valery trace 

participation not only in Emma’s gestures or Lia’s talk but in the interaction among utterances, 

gestures and the numbers on the board. 

Reassigning saliency to multiple semiotic resources contributed to semiotic innovation. 

In Emi’s example, Valery assigned saliency to gestures and movement by quickly bringing the 

class together around Emi’s strategy. Assigning saliency to gestures and movements helped the 

class recognize Emi’s strategy as a valid way to approach the task. How Emi participated in this 

activity influenced how the group participated. This included students, as they took up and 

adapted Emi’s strategy. It also included Valery’s participation, as she did not mandate a specific 

 

94 

procedure to come up with an answer. She worked with students to understand the multiple ways 

and resources they used to make sense of the fractions. 

Valery learned to see how students who tended to be quiet in class were indeed 

generating multiple strategies and contributing ideas by using multiple semiotic resources. She, 

however, expressed a tension, as she questioned whether she was allowing students to remain 

silent, denying them opportunities to learn to talk about their mathematical ideas. For example, 

in Emi’s case, Valery wondered if she should have facilitated a whole class discussion to give 

students opportunities to articulate ideas about fractions related to Emi’s strategy. Valery’s 

awareness of the culture of the school influenced this concern. She thought that in other grades 

teachers may not recognize and validate non-spoken participation, misinterpreting silence as lack 

of interest or understanding. Simultaneously, Valery thought that forcing students who tended to 

be quiet to speak up could be intimidating and off-putting. 

Fluid Participation in Mathematical Activity 

Valery came to value a flexible structure of mathematical activity where students 

developed diverse strategies, coordinated materials, and engaged in fluid ways of participating to 

explore different tasks. An example of this transformation comes from the third intervention 

cycle in the geometry unit. The class spent a few lessons exploring whether circles had perimeter 

and area, eventually agreeing that they do. In one of the lessons, two student strategies to figure 

out the area of a circle emerged and motivated Valery to try a flexible structure for a task related 

to those strategies. I describe the two student strategies first and then I discuss the flexible tasks 

that followed. 

While Valery was launching a task to generate strategies to establish the perimeter of a 

circle, Juan grabbed a piece of graph paper and started to draw a circle that he then erased (figure 

 

95 

6.2a). He grabbed a compass and drew several circles on other sheets (figure 6.2b, 6.2c and 

6.2d). He paused and observed the last two circles he drew (shown in figure 6.2d). Putting away 

the other sheets, Juan grabbed a new sheet, looked back and forth between the circles and the 

new sheet, and finally drew the circles in Figure 6.2e. 

Figure 6.2 Juan’s circles 

 

 

a. 

 

 

b. 

c. 

 

d. 

 

e. 

 

96 

Noticing what Juan was doing, Valery approached him after launching the task: 

Table 6.6 Juan's strategy for the area of a circle 

 

Utterances 

Translation 

Actions 

Valery 

Deja ver, Juan. ¿Qué 

Let me see, Juan. What 

Leaning over, moves paper 

 

hiciste? 

did you do? 

toward her and observes. 

Juan 

Hizo círculos. 

I made circles. 

Looking at the paper. 

Valery 

¿Querías hacer círculos? 

You wanted to make 

Looking at Juan. 

 

 

circles? 

Juan 

Es más o menos una idea. 

That’s more or less the 

 

 

 

 

Quiere ver la área de esto. 

idea. I want to see the 

Points at circles. 

 

area of this.  

 

Valery 

Ajá. Queremos ver cómo 

Uh huh. We want to see 

Moving papers back 

 saber el área y el 

 how to figure out the area 

 toward Juan. Stands back 

perímetro de un círculo. 

and perimeter of a circle. 

up. 

 

 

 

Muy bien. 

Juan 

Es como así. 

 

 

 

 

Very well. 

It’s like this. 

 

 

 

Gestures as if inscribing 

one of the circles in a 

square his hands formed. 

Valery 

¡Ahh! ¡Qué interesante! 

Oh! How interesting! 

 

Instead of asking Juan to put away the graph paper and pay attention to the launching of 

the task, Valery approached him willing to interpret what he was doing as valid participation in 

mathematical activity. She was tracing Juan’s participation in what could have been 

 

97 

misconstrued as off-task behavior. In the excerpt above, Valery initially interpreted Juan’s 

spoken explanations and the drawings as indicating his thinking about the area and perimeter of 

circles, which Valery validated, stating that was what the class was working on. When Juan 

gestured inscribing the circle in a square, Valery understood he was trying to relate the area of 

the circle to the area of the square. Making sense of Juan’s idea involved the interaction of 

multisemiotic resources, including the circles, the gestures, and the spoken language. 

Later in the lesson, Valery assigned saliency to Juan’s idea. Juan went back to his group 

to work on the task generating strategies to figure out the perimeter of a circle. This activity 

included the group ambassadors strategy, where a representative of each group visited and 

helped or got help from another group. When Juan’s group was hosting an ambassador, Valery 

asked Juan to share his circles on the graph paper, writing “para el área” [for the area] as a title. 

Juan used a red marker to trace over the square that inscribed one of the circles. After the 

ambassadors reported back to their original groups, students briefly shared what they found 

interesting, as a whole class. Paige was one of the students who visited Juan’s group. Stacey, 

who heard about Juan’s idea through Paige, commented: 

Me gustó como lo hizo el otro grupo que Paige estuvo porque el área del círculo es así 

casi como área de este cuadrado [levanta y señala el papel donde Paige explicó la 

estrategia de Juan]. Esto es un estimación. Es un buen estimación. 

I liked how the other group that Paige was visiting did it because the area of the circle is 

almost like the area of this square [holds up and points to the paper where Paige 

explained Juan’s strategy]. It’s an estimation. It’s a good estimation. 

In this case, Valery used the ambassadors’ strategy to assign saliency to Juan’s work that 

coordinated the circle, the inscribing square, the smaller squares making up the inscribing square, 

 

98 

and the title “for the area.” Through collaboration among groups, Paige and Stacey became fluid 

participants in this multisemiotic process, understanding Juan’s strategy as a “good estimation” 

to calculate the area of a circle. Rather than focusing on participation occurring in present time 

during a discussion or small group work, Valery facilitated students’ interaction with 

participation initiated in the past, thus making Juan’s circles influence others’ thinking. 

Halley generated the second strategy that the class explored. Responding to Stacey’s 

comment, Halley suggested an alternative. She went to the board and drew a square and a circle 

next to it: 

Table 6.7 Halley's strategy for the area of a circle 

 

Utterances 

Translation 

Actions 

Halley 

Si tienes como algo, como 

If you have like something, like 

 

 

 

 

 

 

 

 

 

slime* aquí, adentro de este 

slime here, inside this circle,  

Points at the 

círculo,  

you can put it inside this square, 

circle. 

puedes ponerlo adentro de este 

 

Points at the 

cuadrado, 

and find the area here. 

square. 

y encontrar el área aquí. 

 

Points at the 

 

And it’s still, it’ll be the same, 

circle. 

Y todavía es, va a ser el mismo, 

how do you say amount? 

¿cómo se dice amount? 

Amount. 

Valery 

Cantidad. 

Amount of slime. 

Halley 

Cantidad de slime. 

*Halley refers to the elastic, gooey arts and crafts material. In this class, students made slime 

using glue, water, paint and glitter. 

 

99 

Halley added to Juan’s idea, using spoken language, the drawings of a circle and a 

square, and the imagined slime she mentioned. Halley typically felt comfortable speaking in 

whole class discussions and articulating her ideas. In the past, her ideas frequently influenced the 

strategies that Valery asked the entire class to use. This time, however, the class showed interest 

in both Juan’s and Halley’s strategies.  

The class discussed both Juan’s and Halley’s ideas and different students wanted to try 

their strategies. In the following lesson, Valery asked students to form groups according to the 

strategy they wanted to use. She provided graph paper with a circle on it. Additionally, the 

groups that wanted to try Halley’s strategy received sugar to cover the circle. Figure 6.3a shows 

sample work from a group that tried Halley’s strategy. These students used a ruler to make sure 

the sugar was uniformly distributed along the circle and to rearrange the sugar in the squares. 

They counted the number of squares that the sugar covered to determine the area of the circle. 

Figure 6.3b shows sample work from a group that tried Juan’s strategy. This group counted the 

number of squares the circle covered, approximating the corners of the inscribing figure as half a 

square. They added fractions (half squares) and whole squares to figure out the area of the circle. 

Figure 6.3 Figuring out the area of a circle 

a. 

 

 

100 

 

b. 

In the gallery walk that followed, students used sticky notes to comment on the two 

strategies. Not only did students comment on the strategies, but they also reacted to each other’s 

comments. For example, reacting to the sugar’s strategy, a student wrote “¡Oh, brillante! Cuando 

necesitas saber el área de círculo, sólo necesitas ir por azúcar. ¡Que una estrategia más eficiente! 

[Oh, brilliant! When you need to find out the area of a circle, all you need to do is get some 

sugar. What an efficient strategy!]. Another student drew an arrow from this note to another that 

read: “Pero esto es exacto, no estimación” [But this is exact, not an estimation]. Similarly, 

reacting to Juan’s inscribing square strategy, a student commented: “Me gusta” [I like it] and put 

a sticker with a thumbs up. Another student placed a sticky note overlapping, commenting: “Esto 

toma para siempre sumando todas las fracciones” [This takes forever, adding all the fractions]. 

When co-analyzing this lesson, Valery focused on the flexible structure of the 

mathematical activity, where different students tried different materials and approaches to 

engage with the idea of area of a circle. Valery mentioned “para que puedan participar tiene que 

haber variedad” [for them to be able to participate, there has to be variety], emphasizing the 

word “variety” by opening up both her arms and hands, as if releasing something she was 

previously holding. Valery recognized that in this lesson there were different ways of 

participating that were beyond her control. She mentioned she could have organized this lesson 

as two tasks, one with all students trying Halley’s strategy and one with all students trying 

Juan’s. Thinking that this arrangement would replicate the kind of repetitive exploration that 

students criticized in the focus groups, Valery decided to integrate both strategies in one task, 

with students choosing which strategy to explore. Valery used a lesson that was driven by 

students’ ideas and interests, instead of following predetermined topics and rigid lesson plans. 

 

101 

She flexibly adjusted the lesson to allow students to explore ideas they were interested in. This 

type of lesson and activities facilitated students’ fluid participation in mathematical activity. 

Rather than using a class discussion for students to explain their work and converge to an 

answer or an assessment of the strategies’ efficiency, Valery used a gallery walk where opinions, 

arguments and counterarguments coexisted around students’ displayed work. As Valery 

commented: “es como si las cosas tan variadas que hicieron de maneras tan diferentes estuvieran 

hablando por ellos. Mejor que eso. Más claro” [It’s as if the things so diverse that they did, in 

such different ways, were speaking in their behalf. Even better than that. More clear]. Having 

come a long way from her initial perspectives on participation as individual talk, Valery now 

seemed attuned to fluid and generative participation, where students, materials, and ideas 

interacted to support meaning-making. Engaged in this hermeneutic listening for participation, 

the students and Valery in interaction seemed to have developed answers to her earlier questions 

about how to bring all students’ ideas into a meaningful dialogue of mutual influence. 

Students noticed the fluidity of participation that the class was moving toward and they 

explicitly suggested moments when they wanted groups to work in different ways. Earlier in the 

geometry unit, for example, students tried different strategies and materials to figure out the 

perimeter of circles. These strategies included using yarn to cover the circumference of a circle 

and then stretching and measuring the yarn, as well as arranging popsicle sticks on the 

circumference to figure out the sum of the lengths of all the sticks. One of the students, Harper, 

referred to a lesson in the fractions unit to suggest that different groups should explore these two 

strategies. When the class was discussing both the yarn and the stick strategies to decide which 

one to use, Harper mentioned they could try both, in a way similar to when they had used unifix 

cubes and paper and scissors to add fractions. This suggestion, along with the multiple evidence 

 

102 

presented in this study, indicates how students rightfully appropriated the transformed culture of 

participation. It was a rightful appropriation because they actively contributed to this 

transformation. 

Valery expressed tensions between the flexible structure of activities and the school 

demands. As the end-of-year student assessment approached, the school administration wanted 

to increase the number of hours that students on Individualized Educational Plans (IEPs) spent 

outside the mainstream classroom working with the IEP teacher aide. Valery’s students would 

join students from the other Spanish immersion third-grade classroom. The school administration 

wanted both teachers to synchronize the textbook pages they studied each day. This school 

expectation prompted Valery to completely devote some lessons to topics and problems from the 

textbook. This strict structure was in stark contrast with the flexibility that she and her students 

had built around their teaching and learning and that allowed them to engage in fluid ways of 

participating. Valery explicitly discussed with the students that some days they would 

exclusively work on the textbook, explaining that it was only in preparation for the end-of-year 

test. Valery was determined to protect the space for diverse participation that we—the students, 

Valery, and I—had forged together. 

 

103 

Chapter 7  Discussion: What It Took to Transform the Culture of Participation 

When Valery and I began discussing participation in this mathematics classroom, we saw 

what seemed to us as a well-defined problem: How to level student talk. Listening carefully to 

students, however, we realized we needed to understand what participation meant and for whom. 

Over time, we went from asking why some students were not participating to asking how they 

were participating. This was a question that neither of us alone could answer and it required 

collaborating with students. In this chapter, I focus on two lessons that Valery and I learned 

through this collaboration that informed transformations in the culture of participation: (1) 

Reconsidering what counts as participation, and (2) approaching participation hermeneutically. I 

make connections to previous literature on participation in mathematics classrooms as a way to 

better understand our learning about participation. The chapter ends with a discussion regarding 

implications for research and practice. 

Reconsidering What Counts as Participation 

To think about participation in generative and inclusive ways, we need to first unlearn 

reductionist and exclusionary views on participation. Initially, the normalization of traditional 

views on participation hindered our ability to see alternatives, even though those alternatives 

were hidden in plain sight. As a cultural group, a classroom is a place for a teacher to raise to 

awareness taken for granted cultural meanings, values, and practices as a first step to question 

and potentially transform them (Louie, 2017; Paris, 2012). In this case, bringing the students’ and 

the teacher’s perspectives into dialogue helped the teacher reinterpret her own perspectives on 

participation. Informed by a participatory research methodology, the teacher and I assumed the 

role of learners, supporting each other in making sense of what participation meant in this 

classroom. 

 

104 

Learning to see beyond quantity to focus on qualities of participation enabled the teacher 

to recognize the important role that social interactions, mathematical tasks and teaching 

strategies play in how students experience participation. Confronting unintended yet problematic 

consequences that the culture of participation had on students’ participation motivated the 

teacher to reimagine her practice in light of the situated meanings of participation in this 

classroom. These meanings of participation that emerged from the dialogue among different 

perspectives serve to add nuances to the views on participation prevalent in the mathematics 

education literature. Contrary to previous studies that have foregrounded the role that individual 

preferences or characteristics play in how students participate (see, for example, Lubienski, 

2000), for these students the type of social interactions influenced their participation. 

Collaborative, fluid interactions where students experienced a sense of mutuality supported 

participation in this classroom. 

Rather than taking on a single form, this fluid participation emerged as observation, 

listening, thinking, helping, drawing, movement and talk. Students infused mathematical activity 

with these ways of participating. This means that to learn to see participation differently, teachers 

and researchers need to listen and observe in moments and places that have not been associated 

with participation, including for example, private gestures, that is, gestures related to 

mathematical activity that students intended for themselves and not for an external audience. It 

also means attending not only to individual instances of participation but also to the effect that 

participation has on other students’ thinking and on the unfolding of the lessons. Learning to see 

participation in this fluid way involved expanding where and when we look for participation, as 

it extends beyond single utterances throughout the span of meaningful mathematical activity. 

 

105 

Unlike many studies that see the teacher’s role in participation in mathematics classrooms 

as equitably distributing student talk (Foote, 2011; Jansen, 2006), in this dissertation the 

teacher’s role involved joining students as a participant with an intense curiosity to learn about 

context-specific ways of participating. Only from this role was the teacher able to challenge her 

unexamined involvement in favoring adult perspectives on participation over those of her 

students. 

Listening Hermeneutically for Participation 

No longer satisfied with looking for predetermined ways of participating, Valery 

wondered what her teaching could look like if she recognized and validated multiple ways of 

participating. Our collaboration helped her connect this reflection with practice as we co-planned 

units. In this planning stage she reimagined and revised the types of activities and teaching 

strategies that could help her see, in situ, the kind of participation she was envisioning. Rather 

than a prescriptive list of steps for the teacher to follow, as she implemented lessons and began to 

recognize participation in practice, she flexibly and responsively drew on her evolving teaching 

repertoire. 

Teaching strategies that previous research has associated with hermeneutic listening 

informed the strategies that Valery brought into her teaching practice, including asking genuine 

questions, directing students’ listening and amplifying participation. Valery not only asked 

genuine questions but she also directed attention to students’ wonderment questions, which 

became a way of participating that allowed for the sense of mutual influence that students 

valued. Valery also provided multiple opportunities for students to amplify each other’s 

participation, including students observing and listening in during gallery walks and as group 

ambassadors. These strategies fostered the type of collaborative interaction that students 

 

106 

mentioned in focus groups, illustrating how teachers can foster social goals that students 

associate with participation (Jansen, 2006). To make sense of and amplify the ingenious and 

multiple ways in which students participated, Valery needed to be responsive to the multiple 

ideas that emerged. Previous research on mathematics class discussions has suggested 

predetermined strategies that “give teachers control over what is likely to happen in a discussion” 

(M. S. Smith, Hughes, Engle, & Stein, 2009, p. 550). In contrast, Valery shared control of what 

happened in discussions and elsewhere with students, responding to their interest in complex 

mathematical ideas and honoring their fluid participation. 

Valery also learned to see both spoken and non-spoken participation as equally valid 

instead of dismissing them as off-task behaviors. Consistent with previous research that has 

reframed off-task behavior as possible participation that can open up resources for mathematical 

meaning-making (Langer-Osuna, 2018), Valery’s learning to listen hermeneutically allowed her 

to interpret these seemingly off-task behaviors differently. These strategies facilitated 

participatory interactions that drew participants—students and the teacher—to learn to listen to 

one another hermeneutically. 

Listening hermeneutically for participation positioned the teacher to recognize 

mathematical ideas in spaces and moments she had not considered before. She revisited and 

revised teaching plans to make room for the flexible exploration of these ideas that interested the 

students. In turn, the tasks she used facilitated the emergence of additional mathematical ideas 

beyond the narrow scope of the textbook topics in unexpected ways of participating that the 

teacher continued to reflect on. This synergistic connection between reflection and practice 

contributed to the development of a dynamic culture of participation influenced by both the 

students and the teacher. 

 

107 

Over time, the culture of participation evolved toward embracing multiple ways of 

participating in cooperative mathematical activity in which students and the teacher addressed 

unexpected ideas that emerged in interactions. Semiotic innovations characterized the 

transformation of the meanings and practices associated with participation, including challenging 

the tradition of studying mathematical ideas prescribed by pacing guides, reassigning saliency to 

non-spoken participation and wonderment questions, and shifting the role of the teacher from an 

equalizer of student talk to a participant joining students in their fluid and dynamic 

understanding of participation. Although this culture of participation conflicted with some school 

demands, the teacher’s excitement about the ways of participating that she was learning to see 

and students’ responsiveness to transformations in her teaching outweighed the influence that 

these restrictive school demands had on the culture of participation. 

Valery fostered a dynamic and inclusive culture of participation. As she learned to see 

participation, students appropriated strategies for listening hermeneutically to what peers had to 

say in words or otherwise. This evolving culture of participation enhanced the opportunities that 

the students and the teacher had to develop the often neglected need for a sense of belonging and 

mutual influence that is at the core of human activity, including mathematical activity. 

Expanding Research on Participation 

The culture of exclusion in mathematics education has been described as the 

marginalization of students who do not align with the dominant culture of mathematics education 

(Louie, 2017). The tendency in the literature to conflate participation with student talk may 

unintentionally contribute to this culture of exclusion by focusing on talkative students. The 

conceptualization of participation as a complex social phenomenon that informed this 

dissertation allowed me to consider participation beyond student talk, including gestures, 

 

108 

observation, listening, and writing. Bringing this conceptualization into dialogue with 

hermeneutic listening informed the focus on learning how students participate in a specific 

culture of participation, as opposed to perpetuating marked and unmarked participation by 

looking at individual student characteristics. 

The focus on the culture of participation highlighted the role that the classroom as a 

cultural group played in participation. Recognizing that what constitutes participation varies 

across and within classrooms brought up questions about what counts as participation and who 

participates in the making of that decision. Findings from this dissertation highlight the 

possibility of exploring through innovative research designs the situated meanings of 

participation in different contexts, using methodologies that elicit both teachers’ and students’ 

meanings of participation. In this dissertation, I used Participatory Design Research (PDR) with 

children as a methodology that guided this exploration. PDR’s iterative process of designing, 

implementing and reflecting on co-designed interventions provided multiple opportunities for the 

teacher to connect her reflection on participation with her practice. PDR also provided multiple 

opportunities to co-create spaces for the teacher to engage in hermeneutic listening with her 

students in order to transform the culture of participation. Future research could draw on 

participatory and inclusive methodologies to foster collaborations among multiple participants 

who influence participation in mathematics classrooms, including, students, teachers, parents, 

and school administrators. 

The social semiotics analytical framework that informed this dissertation drew attention 

to how social interactions and social contexts influence the culture of participation. By 

illustrating some context-specific tensions, this dissertation extends previous research that argues 

that tensions between the classroom and the school play a role in how students participate 

 

109 

(Brown, 2017). In this case, the teacher navigated assessment practices that demanded attention 

to narrow mathematical ideas that often conflicted with the more connected mathematical ideas 

that interested students. Assessment practices also reinforced the authority that predetermined 

pacing guides exerted on participation. These contextual influences that attempted to counteract 

semiotic innovation mediated the teacher’s engagement with hermeneutic listening for 

participation. These findings suggest that future research on hermeneutic listening for 

participation could attend to and address contextual factors that mediate how teachers connect 

their emerging reinterpretation of participation with their practice. 

Reimagining the Teacher as a Learner of Student Participation 

Engaging in hermeneutic listening for participation involved a transformation in the 

teacher role from an observer monitoring and disrupting silence to a participant joining students 

to learn to reinterpret their participation. Willingness to elicit and carefully consider students’ 

perspectives on their own participation could help teachers learn about their students’ ways of 

participating and classroom dynamics that influence or stifle their participation. Instead of 

attempting to bring students to an idealized standard of student talk, a more productive and 

generative approach could be to promote a classroom culture of participation that honors 

students’ multiple ways of participating.  

Rather than prescriptive steps to follow, in this dissertation the connections between 

reflection and practice were dynamic. These connections were informed by our teacher-

researcher collaboration where we planned tentative focal strategies intended to enable and 

amplify multiple and unanticipated ways of participating. By attending to non-spoken 

participation and bringing it to the attention of the class, the teacher validated these contributions 

as important for the learning of mathematical concepts by the whole group. Pedagogical 

 

110 

approaches to help teachers expand and flexibly draw on a teaching repertoire that responsively 

addresses multiple ways of participation could strengthen the connections between teachers’ 

reflection on participation and their practice. 

Tensions between students’ perspectives on participation and Valery’s beliefs regarding 

her role in supporting participation posed a challenge as she initially tried to connect reflection 

and practice. These tensions included how to spotlight ideas in off-task behavior without 

embarrassing students, and how to accentuate students’ non-spoken participation while 

simultaneously providing opportunities to talk about their mathematical ideas. Teachers could 

engage in exploring similar tensions in order to move their practice to a space where they and 

their students can reinterpret what counts as participation. 

Rather than having students participate in predetermined ways, in this dissertation the 

teacher and I worked together to move our understanding of participation closer to the multiple 

ways in which students experience and perceive participation. It took a collaborative effort to 

unlearn normative views on what it means to participate, to make sense of the richness and 

complexity of students’ perspectives on participation. A broader, more inclusive perspective on 

participation, however, did not make its way into the teacher’s practice smoothly. A prolonged, 

iterative process was necessary to connect the teacher’s reflection to a practice where she learned 

to listen hermeneutically for participation. Many were the moments in which the teacher found 

herself surprised by (re)discovering participation through gestures, movements, observation and 

comments. Many were the spaces—physical, intellectual, and temporal—in which she found 

forms of participation previously unaccessible within the culture of participation in which her 

practice had developed. As we—students, teacher and researcher—came together as co-

 

111 

participants in this study, we worked as a group on shaping the important decision of what counts 

as participation in a mathematics classroom. 

 

 

 

 

 

 

112 

 

 

 

 

 

 

 

 

 

 

 

 

APPENDICES 

 

113 

APPENDIX A—Initial Focus Group Questionnaire 

 

We will be exploring your ideas about participation in math class. Each one of you can 

choose how to express your ideas. You can draw, you can talk or you can write about your ideas 

at any moment. [Paper and pencil provided.] 

 

Let’s talk about participation in math class. Take five minutes to draw, write or think 

about how you participate in math class. 

[After five minutes.] Let’s share your ideas. You can show us what you draw or wrote and 

you can talk about your ideas: 

What does it mean for you to participate in class? 

When do you like to participate in math class? When don’t you like to participate? 

Why do you participate in math class? 

How do you decide when to participate in math class? 

What does your teacher do that helps you participate in math class? 

 

 

 

114 

APPENDIX B—Initial Teacher Interview Questionnaire 

 

For you, what is student participation in math class? 

How does participation matter in math class? 

How do different students participate in math class? 

What do you do to help your students participate in math class? 

How do you respond to students’ participation? 

How does what you know about your students inform how you support participation in 

math class? 

What school guidelines or expectations do you think influence how you address 

participation? 

What examples about discussions with other teachers, with parents or supervisors that 

influenced how you view or respond to participation can you give? 

 

115 

 

 

APPENDIX C—Fractions Task 

 

116 

 

 

 

 

 

 

 

 

 

 

 

 

REFERENCES 

 
 
 

 

117 

 

 

REFERENCES 

 

Aguiar, O. G., Mortimer, E. F., & Scott, P. (2010). Learning from and responding to students' 

questions: The authoritative and dialogic tension. Journal of Research in Science Teaching, 
47(2), 174–193. http://doi.org/10.1002/tea.20315 

Anderson, G. L. (1998). Toward authentic participation: Deconstructing the discourses of 

participatory reforms in education. American Educational Research Journal, 35(4), 571–
603. http://doi.org/10.3102/00028312035004571 

Arcavi, A., & Isoda, M. (2007). Learning to listen: From historical sources to classroom practice. 

Educational Studies in Mathematics, 66(2), 111–129. http://doi.org/10.1007/s10649-006-
9075-8 

Atweh, B., Bleicher, R. E., Cooper, T. J., 1998. (1998). The construction of the social context of 

mathematics classrooms: A sociolinguistic analysis. Journal for Research in Mathematics 
Education, 29(1), 63–82. http://doi.org/10.2307/749718 

Bang, M., & Vossoughi, S. (2016). Participatory design research and educational justice: 

Studying learning and relations within social change making. Cognition and Instruction, 
34(3), 173–193. http://doi.org/10.1080/07370008.2016.1181879 

Baxter, J. A., & Woodward, J. (2001). Effects of reform-based mathematics instruction on low 
achievers in five third-grade classrooms. The Elementary School Journal, 101(5), 529–547. 
http://doi.org/10.1086/499686 

Bell, P. (2004). On the theoretical breadth of design-based research in education. Educational 

Psychologist, 39(4), 243–253. http://doi.org/10.1207/s15326985ep3904_6 

Berg, B. (2004). Qualitative research methods for the social sciences. Boston, MA: Pearson 

Education. 

Bergold, J., & Thomas, S. (2012). Participatory research methods: A methodological approach in 

motion. Historical Social Research/Historische Sozialforschung, 37(4), 191–222. 
http://doi.org/10.2307/41756482 

Bishop, A. J. (1988). Mathematics education in its cultural context. Educational Studies in 

Mathematics, 19(2), 179–191. 

Bishop, A. J. (1991). Mathematical enculturation: A cultural perspective on mathematics 

education. Dordrecht, Netherlands: Kluwer Academic Publishers. 

Boaler, J. (1999). Participation, knowledge and beliefs: A community perspective on 

mathematics learning. Educational Studies in Mathematics, 40(3), 259–281. 

 

118 

Booker, A., & Goldman, S. (2016). Participatory design research as a practice for systemic 

repair: Doing hand-in-hand math research with families. Cognition and Instruction, 34(3), 
222–235. http://doi.org/10.1080/07370008.2016.1179535 

Boyden, J., & Ennew, J. (1997). Children in focus: A manual for participatory research with 

children. Stockholm, Sweden: Save the Children Sweden. 

Breyfogle, M. L., & Williams, L. E. (2008). Designing and implementing worthwhile tasks. 

Teaching Children Mathematics, 15(5), 276–280. 

Brisk, M. E. (2011). Bilingual education. In E. Hinkel (Ed.), Handbook of research in second 

language teaching and learning (Volume II) (pp. 7–24). New York, NY: Routledge. 

Brown, R. (2017). Using collective argumentation to engage students in a primary mathematics 

classroom. Mathematics Education Research Journal, 29(2), 183–199. 
http://doi.org/10.1007/s13394-017-0198-2 

Butman, S. M. (2014). A new twist on collaborative learning. Mathematics Teaching in the 

Middle School, 20(1), 52–57. http://doi.org/10.5951/mathteacmiddscho.20.1.0052 

Chin, C., & Brown, D. E. (2010). Student-generated questions: A meaningful aspect of learning 

in science. International Journal of Science Education, 24(5), 521–549. 
http://doi.org/10.1080/09500690110095249 

Civil, M., & Planas, N. (2004). Participation in the mathematics classroom: Does every student 

have a voice? For the Learning of Mathematics, 24(1), 7–12. 

Clark, A., & Moss, P. (2011). Listening to young children: The mosaic approach. London: 

National Children’s Bureau and Joseph Rowntree Foundation. 

Cobb, P., & Yackel, E. (1998). A constructivist perspective on the culture of the mathematics 
classroom. In F. Seeger, J. Voigt, & U. Waschescio (Eds.), The culture of the mathematics 
classroom (pp. 158–190). Cambridge, UK: Cambridge University Press. 

Cohen, E. G. (1994). Restructuring the classroom: Conditions for productive small groups. 

Review of Educational Research, 64(1), 1–35. http://doi.org/10.3102/00346543064001001 

Common Core State Standards Initiative. (2010). Common Core State Standards for 

Mathematics. Washington, DC: National Governors Association Center for Best Practices 
and the Council of Chief State School Officers. Retrieved from 
http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf 

Cook-Sather, A. (2002). Authorizing students' perspectives: Toward trust, dialogue, and change 

in education. Educational Researcher, 31(4), 3–14. 

Cornwall, A., & Jewkes, R. (1995). What is participatory research? Social Science and Medicine, 

41(12), 1667–1676. 

 

119 

Crespo, S. (2000). Seeing more than right and wrong answers: Prospective teachers’ 

interpretations of students’ mathematical work. Journal of Mathematics Teacher Education, 
3, 155–181. 

Creswell, J. W., & Miller, D. L. (2000). Determining validity in qualitative inquiry. Theory Into 

Practice, 39(3), 124–130. http://doi.org/10.1207/s15430421tip3903_2 

Dallimore, E. J., Hertenstein, J. H., & Platt, M. B. (2004). Classroom participation and 

discussion effectiveness: Student-generated strategies. Communication Education, 53(1), 
103–115. http://doi.org/10.1080/0363452032000135805 

Davis, B. (1996). Teaching mathematics: Toward a sound alternative. New York, NY: 

Routledge. 

Davis, B. (1997). Listening for differences: An evolving conception of mathematics teaching. 

Journal for Research in Mathematics Education, 28(3), 355–376. 

Davitti, E., & Pasquandrea, S. (2017). Embodied participation: What multimodal analysis can 

tell us about interpreter-mediated encounters in pedagogical settings. Journal of Pragmatics, 
107, 105–128. http://doi.org/10.1016/j.pragma.2016.04.008 

de Freitas, E., & Zolkower, B. (2009). Using social semiotics to prepare mathematics teachers to 

teach for social justice. Journal of Mathematics Teacher Education, 12(3), 187–203. 

de Freitas, E., & Zolkower, B. (2011). Developing teacher capacity to explore non-routine 
problems through a focus on the social semiotics of mathematics classroom discourse. 
Research in Mathematics Education, 13(3), 229–247. 
http://doi.org/10.1080/14794802.2011.624705 

Doerr, H. M. (2006). Examining the tasks of teaching when using students' mathematical 

thinking. Educational Studies in Mathematics, 62(1), 3–24. http://doi.org/10.1007/s10649-
006-4437-9 

Dominguez, H. (2016). Mirrors and windows into student noticing. Teaching Children 

Mathematics, 22(6), 358–365. http://doi.org/10.5951/teacchilmath.22.6.0358 

Ehret, C., & Hollett, T. (2016). Affective dimensions of participatory design research in informal 

learning environments: Placemaking, belonging, and correspondence. Cognition and 
Instruction, 34(3), 250–258. http://doi.org/10.1080/07370008.2016.1169815 

Emanuelsson, J., & Sahlström, F. (2008). The price of participation: Teacher control versus 

student participation in classroom interaction. Scandinavian Journal of Educational 
Research, 52(2), 205–223. http://doi.org/10.1080/00313830801915853 

Engeström, Y. (2011). From design experiments to formative interventions. Theory & 

Psychology, 21(5), 598–628. http://doi.org/10.1177/0959354311419252 

 

120 

Esmonde, I., & Langer-Osuna, J. M. (2013). Power in numbers: Student participation in 

mathematical discussions in heterogeneous spaces. Journal for Research in Mathematics 
Education, 44(1), 288–315. http://doi.org/10.5951/jresematheduc.44.1.0288 

Fals-Borda, O. (1987). The application of participatory action-research in Latin America. 

International Sociology, 2(4), 329–347. 

Fassinger, P. A. (1995). Understanding classroom interaction: Students' and professors‘ 
contributions to students’ silence. The Journal of Higher Education, 66(1), 82–96. 
http://doi.org/10.1080/uhej20.v066.i01;subPage:string:Access 

Featherstone, H., Crespo, S., Jilk, L. M., Oslund, J. A., Parks, A. N., & Wood, M. B. (2011). 

Smarter together! Collaboration and equity in the elementary math classroom. Reston, VA: 
National Council of Teachers of Mathematics. 

Foote, M. Q., & Bartell, T. G. (2011). Pathways to equity in mathematics education: How life 

experiences impact researcher positionality. Educational Studies in Mathematics, 78(1), 45–
68. http://doi.org/10.1007/s10649-011-9309-2 

Foote, M. Q., & Lambert, R. (2011). I have a solution to share: Learning through equitable 

participation in a mathematics classroom. Canadian Journal of Science, Mathematics and 
Technology Education, 11(3), 247–260. http://doi.org/10.1080/14926156.2011.595882 

Franke, M. L., Turrou, A. C., Webb, N. M., Ing, M., Wong, J., Shin, N., & Fernandez, C. H. 

(2015). Student engagement with others' mathematical ideas: The role of teacher invitation 
and support moves. The Elementary School Journal, 116(1), 126–148. 
http://doi.org/10.1086/683174 

Gearhart, M., & Saxe, G. B. (2014). Differentiated instruction in shared mathematical contexts. 

Teaching Children Mathematics, 20(7), 426–435. 
http://doi.org/10.5951/teacchilmath.20.7.0426 

Ghousseini, H., Lord, S., & Cardon, A. (2017). Supporting math talk in small groups. Teaching 

Children Mathematics, 23(7), 422–428. http://doi.org/10.5951/teacchilmath.23.7.0422 

Glesne, C. (2011). Becoming qualitative researchers: An introduction. Boston, MA: Pearson. 

Goldin, S., O'Neill, M. K., Naik, S. S., & Gomez Zaccarelli, F. (2019). Supporting students' 

learning practices: Redefining participation and engagement. The Elementary School 
Journal, 119(3), 417–442. http://doi.org/10.1086/701654 

Gresalfi, M., Martin, T., Hand, V., & Greeno, J. (2009). Constructing competence: An analysis 

of student participation in the activity systems of mathematics classrooms. Educational 
Studies in Mathematics, 70(1), 49–70. http://doi.org/10.1007/sl0649-008-9141-5 

Groundwater-Smith, S., Dockett, S., & Bottrell, D. (2014). Innovative methods. In S. 

Groundwater-Smith, S. Dockett, & D. Bottrell (Eds.), Participatory research with children 

 

121 

and young people (pp. 1–32). London: SAGE Publications. 
http://doi.org/10.4135/9781473910751 

Gutierrez, K. D., Engeström, Y., & Sannino, A. (2016). Expanding Educational Research and 

Interventionist Methodologies. Cognition and Instruction, 34(3), 275–284. 
http://doi.org/10.1080/07370008.2016.1183347 

Halliday, M. A. K. (1978). Language as social semiotic: The social interpretation of language 

and meaning. London: Edward Arnold. 

Hammel, J., Magasi, S., Heinemann, A., Whiteneck, G., Bogner, J., & Rodriguez, E. (2009). 

What does participation mean? An insider perspective from people with disabilities. 
Disability and Rehabilitation, 30(19), 1445–1460. 
http://doi.org/10.1080/09638280701625534 

Hand, V., Kirtley, K., & Matassa, M. (2015). Narrowing participation gaps. The Mathematics 

Teacher, 109(4), 262–268. http://doi.org/10.5951/mathteacher.109.4.0262 

Hansen, H. P., Ramstead, J., Richer, S., Smith, S., & Stratton, M. (2001). Unpacking 

participatory research in education. Interchange, 32(2), 295–322. 

Hardy, T. (2007). Participation and performance: Keys to confident learning in mathematics? 

Research in Mathematics Education, 9(1), 21–32. 
http://doi.org/10.1080/14794800008520168 

Hart, S., Dixon, A., Drummond, M. J., & McIntyre, D. (2004). Learning without limits. New 

York, NY: Open University Press. 

Hickey, D. T. (2003). Engaged participation versus marginal nonparticipation: A stridently 

sociocultural approach to achievement motivation. The Elementary School Journal, 103(4), 
401–429. http://doi.org/10.1086/499733 

Hill, M. (1997). Participatory research with children. Child and Family Social Work, 2(3), 171–

183. http://doi.org/10.1046/j.1365-2206.1997.00056.x 

Hintz, A., & Tyson, K. (2015). Complex listening: Supporting students to listen as mathematical 

sense-makers. Mathematical Thinking and Learning, 17(4), 296–326. 
http://doi.org/10.1080/10986065.2015.1084850 

Hodge, L. L., & Cobb, P. (2016). Two views of culture and their implications for mathematics 

teaching and learning. Urban Education. Advance online publication. 
http://doi.org/10.1177/0042085916641173 

Hoffmann, M. H. G. (2006). What is a “semiotic perspective,” and what could it be? Some 

comments on the contributions to this special issue. Educational Studies in Mathematics, 
61(1-2), 279–291. http://doi.org/10.1007/s10649-006-1456-5 

 

122 

Ing, M., Webb, N. M., Franke, M. L., Turrou, A. C., Wong, J., Shin, N., & Fernandez, C. H. 

(2015). Student participation in elementary mathematics classrooms: The missing link 
between teacher practices and student achievement? Educational Studies in Mathematics, 
90(3), 341–356. http://doi.org/10.1007/s10649-015-9625-z 

Jacquez, F., Vaughn, L. M., & Wagner, E. (2013). Youth as partners, participants or passive 

recipients: A review of children and adolescents in community‐based participatory research 
(CBPR). American Journal of Community Psychology, 51(1-2), 176–189. 
http://doi.org/10.1007/s10464-012-9533-7 

Jansen, A. (2006). Seventh graders' motivations for participating in two discussion-oriented 

mathematics classrooms. The Elementary School Journal, 106(5), 409–428. 
http://doi.org/10.1086/505438 

Johnson, E. M. S., & Larsen, S. P. (2012). Teacher listening: The role of knowledge of content 

and students. Journal of Mathematical Behavior, 31(1), 117–129. 
http://doi.org/10.1016/j.jmathb.2011.07.003 

Kane, R. G., & Maw, N. (2005). Making sense of learning at secondary school: Involving 

students to improve teaching practice. Cambridge Journal of Education, 35(3), 311–322. 
http://doi.org/10.1080/03057640500319024 

Kirk, S. (2007). Methodological and ethical issues in conducting qualitative research with 

children and young people: A literature review. International Journal of Nursing Studies, 
44(7), 1250–1260. http://doi.org/10.1016/j.ijnurstu.2006.08.015 

Krummheuer, G. (2007). Argumentation and participation in the primary mathematics 

classroom. Journal of Mathematical Behavior, 26(1), 60–82. 
http://doi.org/10.1016/j.jmathb.2007.02.001 

Lack, B., Swars, S. L., & Meyers, B. (2014). Low-and high-achieving sixth-grade students' 
access to participation during mathematics discourse. The Elementary School Journal, 
115(1), 97–123. http://doi.org/10.1086/676947 

Ladson-Billings, G. (2014). Culturally relevant pedagogy 2.0: a.k.a. The remix. Harvard 

Educational Review, 84(1), 74–84. 

Langer-Osuna, J. (2018). Productive disruptions: Rethinking the role of off-task interactions in 

collaborative mathematics learning. Education Sciences, 8(2), 87-97. 
http://doi.org/10.3390/educsci8020087 

Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. 

Cambridge, England: Cambridge University Press. 

Lo, J.-J., Wheatley, G. H., & Smith, A. C. (1994). The participation, beliefs, and development of 

arithmetic meaning of a third-grade student in mathematics class discussions. Journal for 
Research in Mathematics Education, 25(1), 30–49. 

 

123 

Louie, N. L. (2017). The culture of exclusion in mathematics education and its persistence in 

equity-oriented teaching. Journal for Research in Mathematics Education, 48(5), 488–519. 
http://doi.org/10.5951/jresematheduc.48.5.0488 

Lubienski, S. T. (2000). A clash of social class cultures? Students' experiences in a discussion-
intensive seventh- grade mathematics classroom. The Elementary School Journal, 100(4), 
377–403. 

Mack, N. K. (2007). Gaining insights into children's geometric knowledge. Teaching Children 

Mathematics, 14(4), 238–245. 

Mafra Goulart, M. I., & Roth, W. M. (2006). Margin|centre: Toward a dialectic view of 

participation. Journal of Curriculum Studies, 38(6), 679–700. 
http://doi.org/10.1080/00220270600692936 

Matthews, W. (1984). Influences on the learning and participation of minorities in mathematics. 

Journal for Research in Mathematics Education, 15(2), 84–95. 
http://doi.org/10.2307/748884 

Moguel, D. (2004). What does it mean to participate in class?: Integrity and inconsistency in 

classroom interaction. The Journal of Classroom Interaction, 39(1), 19–29. 

Morgan, C. (2006). What does social semiotics have to offer mathematics education research? 
Educational Studies in Mathematics, 61(1-2), 219–245. http://doi.org/10.1007/s10649-006-
5477-x 

Moschkovich, J. (2018). Considering silence and precision: Communication practices that 
support equitable mathematical discussions for marginalized students. In R. Hunter, M. 
Civil, B. Herbel-Eisenmann, N. Planas, & D. Wagner (Eds.), Mathematical discourse that 
breaks barriers and creates space for marginalized learners (pp. 61–83). Rotterdam, 
Netherlands: Sense Publishers. 

Nasir, N. S., Hand, V., & Taylor, E. V. (2008). Culture and mathematics in school: Boundaries 

between "cultural" and “domain” knowledge in the mathematics classroom and beyond. 
Review of Research in Education, 32(1), 187–240. 
http://doi.org/10.3102/0091732X07308962 

National Council of Teachers of Mathematics NCTM. (2000). Principles and standards for 

school mathematics. Reston, VA: NCTM. 

Nickson, M. (1994). The culture of the mathematics classroom: an unknown quantity? In S. 

Lerman (Ed.), Cultural perspectives on the mathematics classroom (pp. 7–35). Dordrecht, 
Netherlands: Springer. 

Norton, B. (2001). Non-participation, imagined communities and the language classroom. In M. 
P. Breen (Ed.), Learner contributions to language learning: New directions in research (pp. 
159–171). New York, NY. 

 

124 

O'Connor, C., Michaels, S., Chapin, S., & Harbaugh, A. G. (2017). The silent and the vocal: 
Participation and learning in whole-class discussion. Learning and Instruction, 48, 5–13. 
http://doi.org/10.1016/j.learninstruc.2016.11.003 

O'Halloran, K. L. (2005). Mathematical discourse: Language, symbolism and visual images. 

New York, NY: Continuum. 

O'Halloran, K. L. (2015). The language of learning mathematics: A multimodal perspective. 

Journal of Mathematical Behavior, 40, 63–74. http://doi.org/10.1016/j.jmathb.2014.09.002 

Paris, D. (2012). Culturally sustaining pedagogy: A needed change in stance, terminology, and 

practice. Educational Researcher, 41(3), 93–97. 

Paris, D., & Alim, H. S. (2014). What are we seeking to sustain through culturally sustaining 

pedagogy? A loving critique forward. Harvard Educational Review, 84(1), 85–100. 

Parrish, S. (2010). Number talks. Helping children build mental math and computation 

strategies, grades K-5. Sausalito, CA: Math Solutions. 

Pimm, D. (1994). Spoken mathematical classroom culture: Artifice and artificiality. In S. 

Lerman (Ed.), Cultural perspectives on the mathematics classroom (pp. 133–147). 
Dordrecht, Netherlands: Springer. 

Powell, A. B., Jeffries, D. A., & Selby, A. E. (1989). An empowering, participatory research 
model for humanistic mathematics pedagogy. Humanistic Mathematics Network Journal, 
1(4), 29–38. 

Punch, S. (2002). Research with children: The same or different from research with adults? 

Childhood, 9(3), 321–341. http://doi.org/10.1177/0907568202009003005 

Reinholz, D. L., & Shah, N. (2018). Equity analytics: A methodological approach for quantifying 

participation patterns in mathematics classroom discourse. Journal for Research in 
Mathematics Education, 49(2), 140–177. http://doi.org/10.5951/jresematheduc.49.2.0140 

Rocca, K. A. (2010). Student participation in the college classroom: An extended 

multidisciplinary literature review. Communication Education, 59(2), 185–213. 
http://doi.org/10.1080/03634520903505936 

Rodd, M. (2003). Witness as participation: The lecture theatre as site for mathematical awe and 

wonder. For the Learning of Mathematics, 23(1), 15–21. 

Rogoff, B., Paradise, R., Mejía Arauz, R., Correa-Chávez, M., & Angelillo, C. (2003). Firsthand 

learning through intent participation. Annual Review of Psychology, 54(1), 175–203. 
http://doi.org/10.1146/annurev.psych.54.101601.145118 

Rudduck, J. (2007). Student voice, student engagement, and school reform. In D. Thiessen & A. 

Cook-Sather (Eds.), International handbook of student experience in elementary and 
secondary school (pp. 587–610). Dordrecht, Netherlands: Springer. 

 

125 

Sáenz-Ludlow, A., & Presmeg, N. (2006). Guest editorial: Semiotic perspectives on learning 

mathematics and communicating mathematically. Educational Studies in Mathematics, 61(1-
2), 1–10. http://doi.org/10.1007/s10649-005-9001-5 

Schultz, K. (2010). After the blackbird whistles: Listening to silence in classrooms. Teachers 

College Record, 112(11), 2833–2849. 

Skott, J. (2013). Understanding the role of the teacher in emerging classroom practices: 

Searching for patterns of participation. ZDM Mathematics Education, 45(4), 547–559. 
http://doi.org/10.1007/s11858-013-0500-z 

Smith, H., & Higgins, S. (2006). Opening classroom interaction: The importance of feedback. 

Cambridge Journal of Education, 36(4), 485–502. 
http://doi.org/10.1080/03057640601048357 

Smith, M. S., Hughes, E. K., Engle, R. A., & Stein, M. K. (2009). Orchestrating discussions. 

Mathematics Teaching in the Middle School, 14(9), 548–556. 

Steinberg, S. R., & Kincheloe, J. L. (2010). Power, emancipation, and complexity: Employing 

critical theory. Power and Education, 2(2), 140–151. 
http://doi.org/10.2304/power.2010.2.2.140 

Takeuchi, M. (2015). The situated multiliteracies approach to classroom participation: English 
language learners' participation in classroom mathematics practices. Journal of Language, 
Identity, and Education, 14(3), 159–178. http://doi.org/10.1080/15348458.2015.1041341 

Tent, M. W. (2001). Circles and the number π. Mathematics Teaching in the Middle School, 6(8), 

452–455. 

Thomas, J. N. (2018). Talking with our hands. Teaching Children Mathematics, 24(5), 308–314. 

http://doi.org/10.5951/teacchilmath.24.5.0308 

Thomas, N., & O'Kane, C. (1998). The ethics of participatory research with children. Children & 

Society, 12(5), 336–348. http://doi.org/10.1111/j.1099-0860.1998.tb00090.x 

Thomas, R., Whybrow, K., & Scharber, C. (2012a). A conceptual exploration of participation. 
Section I: Introduction and early perspectives. Educational Philosophy and Theory, 44(6), 
594–613. http://doi.org/10.1111/j.1469-5812.2010.00731.x 

Thomas, R., Whybrow, K., & Scharber, C. (2012b). A conceptual exploration of participation. 

Section II: Participation as engagement in experience—An aesthetic perspective. 
Educational Philosophy and Theory, 44(7), 746–759. http://doi.org/10.1111/j.1469-
5812.2010.00739.x 

Thomas, R., Whybrow, K., & Scharber, C. (2012c). A conceptual exploration of participation. 
Section III: Utilitarian perspectives and conclusion. Educational Philosophy and Theory, 
44(8), 801–817. http://doi.org/10.1111/j.1469-5812.2010.00740.x 

 

126 

Turner, E., Dominguez, H., Maldonado, L., & Empson, S. (2013). English learners' participation 

in mathematical discussion: Shifting positionings and dynamic identities. Journal for 
Research in Mathematics Education, 44(1), 199–234. 
http://doi.org/10.5951/jresematheduc.44.1.0199 

Vakil, S., & de Royston, M. M. (2016). Rethinking race and power in design-based research: 

Reflections from the field. Cognition and Instruction, 34(3), 194–209. 
http://doi.org/10.1080/07370008.2016.1169817 

van Leeuwen, T. (2005). Introducing social semiotic. New York, NY: Routledge. 

Vannini, P. (2007). Social semiotics and fieldwork: Methods and analytics. Qualitative Inquiry, 

13(1), 113–140. http://doi.org/10.1177/1077800406295625 

Veale, A. (2015). Creative methodologies in participatory research with children. In S. Greene & 

D. Hogan (Eds.), Researching children's experience (pp. 253–273). London, UK: SAGE 
Publications. 

Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. 

Cambridge, MA: Harvard University Press. 

Wager, A. A. (2014). Noticing children's participation: Insights into teacher positionality toward 

equitable mathematics pedagogy. Journal for Research in Mathematics Education, 45(3), 
312–350. http://doi.org/10.5951/jresematheduc.45.3.0312 

Webb, N. M., Franke, M. L., Ing, M., Turrou, A. C., Johnson, N. C., & Zimmerman, J. (2017). 

Teacher practices that promote productive dialogue and learning in mathematics classrooms. 
International Journal of Educational Research, 1–11. 
http://doi.org/10.1016/j.ijer.2017.07.009 

Webb, N. M., Franke, M. L., Ing, M., Wong, J., Fernandez, C. H., Shin, N., & Turrou, A. C. 

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

Wenger, E. (1998). Communities of practice: Learning, meaning and identity. New York, NY: 

Cambridge University Press. 

Williams-Pierce, C. C., Pier, E. L., Walkington, C., Boncoddo, R., Clinton, V., Alibali, M. W., & 

Nathan, M. J. (2017). What we say and how we do: Action, gesture, and language in 
proving. Journal for Research in Mathematics Education, 48(3), 248–260. 
http://doi.org/10.5951/jresematheduc.48.3.0248 

Wilson, J. (2012). Show me a sign. Teaching Children Mathematics, 19(2), 82–89. 

http://doi.org/10.5951/teacchilmath.19.2.0082 

Wood, M. B., & Hackett, M. (2018). The problem of overparticipation: Fostering productive 

participation in your classroom. In S. Crespo, S. Celedón-Pattichis, & M. Civil (Eds.), 

 

127 

Access and equity: Promoting high-quality mathematics in grades 3-5 (pp. 133–146). 
Reston, VA: NCTM. 

Wood, M. B., Sheldon, J., Felton-Koestler, M. D., Oslund, J., Parks, A. N., Crespo, S., & 

Featherstone, H. (2019). 8 teaching moves supporting equitable participation. Teaching 
Children Mathematics, 25(4), 218–223. http://doi.org/10.5951/teacchilmath.25.4.0218 

Wood, T., Williams, G., & McNeal, B. (2006). Children's mathematical thinking in different 

classroom cultures. Journal for Research in Mathematics Education, 37(3), 222–255. 

Wright, von, M. (2006). The punctual fallacy of participation. Educational Philosophy and 

Theory, 38(2), 159–170. http://doi.org/10.1111/j.1469-5812.2006.00185.x 

 
 

 

128