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.
v
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
59
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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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:
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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.
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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.
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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.
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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
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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
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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
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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.
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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
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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.
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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
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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,
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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
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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.
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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,
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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
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(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
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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.
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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?
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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?
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APPENDIX C—Fractions Task
116
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