Journal of Urban Mathematics Education 

December 2017, Vol. 10, No. 2, pp. 106–139 

©JUME. http://education.gsu.edu/JUME 

 

 

CATHERY YEH is an assistant professor in the College of Educational Studies at Chapman 

University, 1 University Drive, Orange, CA 92866; email: yeh@chapman.edu. Her research examines 

issues of equity in mathematics. In particular, she is interested in capturing teachers’ efforts to disrupt 

language, gender, and dis/ability hierarchies in mathematics classrooms.  

Math Is More Than Numbers: Beginning  

Bilingual Teachers’ Mathematics Teaching 

Practices and Their Opportunities to Learn 
 

Cathery Yeh 

Chapman University 

 
In this article, the author provides results from a 3-year, longitudinal study that ex-

amined two novice bilingual teachers’ mathematics teaching practices and their 

professional opportunities to learn to teach. Primary data sources included vide-

otaped mathematics lessons, teacher interviews, and field notes of their teacher 

preparation methods courses. Findings revealed that the teachers were oriented 

toward differing views of learning that shaped how they organized students’ learn-

ing of language and mathematics during classroom instruction. While both teach-

ers used similar teaching strategies to support students’ development of mathemat-

ics specific literacies, there were variances in how the learners were positioned 

within the classroom community and how and which repertoires of language prac-

tices were available and used during mathematics instruction. The teachers’ differ-

ing orientations toward learning are traced to their own professional opportunities 

to learn to teach. The significance of recognizing both the acquisition and partici-

pation metaphors of learning and the development of linguistically and culturally 

relevant teacher education are discussed. 

 

KEYWORDS: bilingual education, mathematics education, novice teachers 

 

mergent bilinguals comprise the fastest growing demographic group and repre-

sent a fourth of all urban public school students in the United States (García, 

Kleifgen, & Falchi, 2008; Menken, Kleyn, & Chae, 2012). As the population of 

emergent bilinguals grows across the country, the prevailing view about bilingual 

education has become increasingly more negative and debates around language pol-

icy more contentious (see, e.g., Flores & Bale, 2016; Menken et al., 2012; Ovando, 

2000; Sleeter, Neal, & Kumashiro, 2015). Despite 40 years of research on the bene-

fits of bilingual education, the mathematics learning experiences of emergent bilin-

guals have mirrored U.S. language politics of xenophobia and assimilation (Aqui-

no-Sterling, Rodríguez-Valls, & Zahner, 2016; Khisty & Willey, 2008; Valdés, 

2015; Valenzuela, 2002). Emergent bilinguals educated in Western schooling prac-

tices have learned to perform mathematics using algorithms that are not their own, 

in a language different than their native tongue, and solving mathematics problems 

E 

http://education.gsu.edu/JUME
mailto:yeh@chapman.edu


 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  107 

irrelevant to their interests and experiences (Aquino-Sterling et al., 2016; García et 

al., 2008; Moschkovich, 2012).  

In the current educational backlash against public education, and bilingual ed-

ucation in particular, political and financial groups such as the English-only move-

ment are growing and their primary goal is to maintain Euro-American dominance 

(see, e.g., Aquino-Sterling et al., 2016; García et al., 2008; Khisty & Willey, 2008; 

Valdés, 2015; Valenzuela, 2002). Nonetheless, there exists a growing counter-

insurgence of bilingual teachers (Achinstein & Aguirre, 2008; Achinstein & Oga-

wa, 2011; Athanases, Banes, & Wong, 2015; Sleeter et al., 2015; Villegas & Irvine, 

2010). The very nature of bilingual education represents a resistance toward cultur-

al and linguistic hegemony. 

In the research project reported here, I examine the teaching and learning ex-

periences of two socially conscious1 novice bilingual teachers serving in the same 

urban2 elementary school in a state where English-only policies have been codified 

into law. Specifically, the purpose of the 3-year longitudinal study, which followed 

two bilingual teachers from their initial teacher preparation program through their 

first 2 years of teaching, was to gain an understanding of their mathematics teach-

ing practices and their own opportunities to learn to teach. The emphasis of the 

work presented is not only to understand their mathematics teaching in bilingual 

settings but also to understand how their teaching practices are negotiated and 

grounded by lived experiences and the perspectives developed over time and 

through their everyday teaching. Studies that examine beginning teachers’ experi-

ences, especially with bilingual mathematics curricula, are scarce (Ingersoll & 

Strong, 2011; Sleeter et al., 2015) and important in advancing understanding of the 

processes in which teacher preparation programs can increase their potential to pre-

pare critical bilingual educators. 

 
Literature Review 

 

Conceptually, I build the project beginning with the idea that different ways 

of understanding learning contribute to our knowledge of the relationship between 

language and mathematics in bilingual classrooms. Therefore, I consider both cog-

                                                 
1 The term socially conscious refers to the two teachers’ expressed commitment to social justice and 

issues of equity. These teachers entered the profession with the goal to challenge English hegemony 

in schools and to leverage the cultural and linguistic resources of children and their communities.  

  
2 The term urban takes on multiple meanings in public discourse and in educational research. Here, 

urban refers to a place with high population density, and urban schools take on two characteristics. 

First, urban schools most often serve a diversity of students across racial, linguistic, and 

socioeconomic backgrounds (among other characteristics). Second, urban schools are most often part 

of a large school district characterized with bureaucratic leadership structures, emphasis on 

standardized testing, and high teacher turnovers.  



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  108 

nitive and sociocultural theories and pedagogical understandings on bilingualism in 

mathematics education (e.g., Cummins, 1984; Gutiérrez, 2002; Gutiérrez, 

Sengupta-Irving, & Dieckmann, 2010; Halliday, 1978; Moschkovich, 2007, 2012; 

O’Halloran, 2005; Téllez, Moschkovich, & Civil, 2011) and in the critiques and 

theories on the politics of language in the United States (Cummins, 2000; Khisty & 

Willey, 2008; Setati, 2008; Valenzuela, 2002). In the following section, I provide a 

historical account of bilingualism in mathematics education as well as research on 

beginning bilingual teachers of mathematics to provide context for the tensions, 

contradictions, and different motives that shape language and mathematics learning 

in bilingual classrooms.  

  

Language and Mathematics Learning in Bilingual Classrooms 
 

The relationship between language and mathematics in research began in the 

field of psychology and examined the cognitive functioning of bilingual learners 

during arithmetic computations and the solving of word problems (see Moschko-

vich, 2007 for a review of early research). These studies brought to the fore the im-

portance of language in mathematics education. Simultaneously learning a second 

language and mathematics is cognitively demanding and can slow down the process 

of mathematics learning. Language demands, and even the absence of a common 

language, can generate tensions and impact who is (and is not) participating in 

classroom interactions (Cummins, 1984; Gutiérrez et al., 2010; Halliday, 1978; 

Pimm, 1987; Setati, 2008; Spanos, Rhodes, Dale, & Crandall, 1988). Contrary to 

popular beliefs, math is more than just numbers; school mathematics is linguistical-

ly complex, with multiple language modalities—reading, writing, listening, speak-

ing, and representing—in play (Aguirre & Bunch, 2012; Cummins, 1984; Dutro & 

Moran, 2003; Halliday, 1978; Kress, 2010; O’Halloran, 2005). 

Over the last three decades, there has been growing attention in the mathemat-

ics education research community on bilingual and multilingual learners. The ma-

jority of past and current studies have examined the effects of students’ language 

proficiencies on academic performances and how teachers’ facilitation and scaf-

folding can support students’ acquisition of language and mathematics content (e.g., 

Cuevas, 1984; Cummins, 1984; Dutro & Moran, 2003; Halliday, 1978; O’Halloran, 

2005; Pimm, 1987; Quinn, Lee, & Valdés, 2012; Spanos et al., 1988). However, 

language is only one of many semiotic resources (e.g., physical control of space, 

gesture, gaze) at play in bilingual/multilingual classrooms (Aguino-Sterling et al., 

2016; DaSilva Iddings, 2005; Domínguez, 2005; Gutiérrez et al., 2010; Turner, 

Dominquez, Empson, & Maldonaldo, 2013; Zahner & Moschkovich, 2011). 

A growing number of studies demonstrate that language use goes beyond lan-

guage proficiency. In a study of student interactions within a reform-based calculus 

class, Gutiérrez (2002) found bilinguals fluent in English spoke in Spanish not as a 

necessity but as a way to bond with others. Moschkovich’s (2007) study of code 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  109 

switching in mathematical conversations similarly builds on language hybridity. 

Her analysis of student discussions found that code switching between English and 

Spanish, using everyday colloquialisms and school-based discourse, was most 

prevalent when students were negotiating ideas and engaging in mathematical ar-

guments. These studies suggest that language and choice of language, more than 

reflecting proficiency, are integral to the identity work students engage as they 

learn. Bilingual speakers use language not only on the basis of proficiency but also 

as an expression of identity (Cummins, 2000; Khisty & Willey, 2008; Setati, 2008; 

Valenzuela, 2002; White, Crespo, & Civil, 2016). Validation and maintenance of 

students’ linguistic identities are intricately linked to academic performance. Bilin-

guals who can read, write, and communicate in their home language are more likely 

to enroll in advanced mathematics courses and continue to higher education (García 

et al., 2008; Khisty & Willey, 2008). Instruction in a student’s home language and 

its validation play significant roles in achievement for bilingual learners (Aquino-

Sterling et al., 2016; Celedón-Pattichis & Ramirez, 2012). 

The literature reviewed offers important contributions to this study on devel-

oping principled instruction for bilingual mathematics learners. First, teachers must 

facilitate students’ acquisition of both the mathematical knowledge and language 

skills needed to engage in meaningful disciplinary discourse practices (Aguirre & 

Bunch, 2012; Aquino-Sterling et al., 2016; Celedón-Pattichis & Ramirez, 2012; Cum-

mins, 1984; Moschkovich, 2012; O’Halloran, 2005; Pimm, 1987; Quinn et al., 

2012). Second, mathematics education research shows that patterns of classroom 

interaction as it relates to student agency, status, and positioning affect students’ 

access and achievement in the classroom (Gutiérrez, 2002; Khisty & Willey, 2008; 

Turner et al., 2013; Zahner & Moschkovich, 2011). Third, bilingual teachers need 

to leverage the varied linguistic repertoires bilingual students bring into classrooms 

and can employ to learn and do mathematics (Celedón-Pattichis & Ramirez, 2012; 

Domínguez, 2005; Gutiérrez et al., 2010; Téllez et al., 2011; Zahner & Moschko-

vich, 2011). This body of work, encompassing over three decades of research, 

demonstrates the advantages of bilingual education; bilingual education still serves 

as the most divisive approach to teaching the 12 million emergent bilingual students 

in the United States. 

  

The Politics of Bilingual Education in the United States 
  

The question arises, why is there so much controversy over bilingual educa-

tion? Tension in bilingual education involves complex issues of power, identity, 

and social status tied to an American history of xenophobia (Arce, 2004; García et 

al., 2008; Ovando, 2000). As Arce (2004) argues, anti-bilingual movements are 

grounded on racialized discourse that, “to be ‘American’ means using only one lan-

guage English and accepting the dominant culture’s norms and values” (p. 230). It 

is no coincidence that federal government and state efforts toward banning bilingual 



 

 

 

 

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education in California (1998), Arizona (2000), and Massachusetts (2002) occurred 

during the largest wave of non-English speaking immigration in American history. 

The very nature of bilingual education, with the embracing of diverse cultures 

and recognition of other languages as significant to American mainstream institu-

tions, serves as counter-resistance to Euro-American and English hegemony. 

Schools are transmitters for capitalist interests as well as locations for opposition to 

the economic and political interests of the dominant (Giroux, 2001; McLaren & 

Jaramillo, 2007; Monzó & McLaren, 2014). Bilingual educators have the potential 

to break the power structures in schooling by challenging hierarchical mandates and 

leveraging the cultural and linguistic resources of children and their communities. 

  

Bilingual Teachers of Mathematics 
 

The need for well-prepared bilingual teachers is critical, which helps to ex-

plain why a growing body of research has focused on teacher recruitment and the 

practices of experienced bilingual teachers. Studies spanning over two decades in-

dicate that bilingual teachers often enter and remain in teaching driven by a human-

istic commitment to give back to their own communities and to redress inequities in 

schooling for students who share their cultural backgrounds (Achinstein & Aguirre, 

2008; Achinstein & Ogawa, 2011; Cavazos, 2009; Gomez, Rodriquez, & Agosto, 

2008; Sleeter et al., 2015; Villegas & Irvine, 2010). A growing body of scholarship 

suggests that bilingual teachers may be more capable of supporting language learn-

ing, tapping into the cultural resources in themselves and their students, and posi-

tioning bilingual learners as capable contributors to the classroom learning commu-

nity (e.g., Cavazos, 2009; Celedón-Pattichis, Musanti, & Marshall, 2010; Gutiérrez, 

2002; Musanti, Celedón-Pattichis, & Marshall, 2009; Remillard & Cahnmann, 

2005; Sleeter et al., 2015; Turner et al., 2013; Zahner & Moschkovich, 2011). 

While these studies justify teacher diversity initiatives, the larger research commu-

nity currently knows little about the experiences of novice bilingual teachers, par-

ticularly in mathematics, and what actually happens when graduates of teacher 

preparation programs begin professional teaching (Ingersoll & Strong, 2011; Sleet-

er et al., 2015). 

What is known, however, is that the current orientation toward language and 

the structured sets of expectations and accepted teaching practices that dominate 

U.S. schools and teacher education programs have largely perpetuated language 

policies of assimilation (Gomez et al., 2008; Khisty & Willey, 2008; López, Scan-

lan, & Gundrum, 2013; Ovando, 2000; Sleeter et al., 2015; Téllez et al., 2011; 

Valenzuela, 2002; White et al., 2016). The majority of U.S. teachers, including bi-

lingual teachers, received the majority, if not all, of their own K–12 schooling and 

teacher preparation in English with a focus on supporting students to “acquire Eng-

lish” rather than maintaining bilingualism. Mathematics is still often seen as con-

sisting of just numbers; rarely do bilingual teachers have professional development 



 

 

 

 

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opportunities centered on discussions of language and mathematics (Aquino-

Sterling et al., 2016; Celedón-Pattichis et al., 2010; Musanti et al., 2009). 

Sleeter and colleagues (2015) contend that the teacher-education research 

community has concentrated on examining teacher preparation when more attention 

should be placed on what actually happens when graduates of teacher preparation 

programs begin to teach. A close examination of novice bilingual teaching in relation 

to teacher preparation is critical for school leaders and teacher educators to consider 

in our work to better support our teacher workforce and our students.  

 
Methods 

 

Purpose and Research Questions 
 

In the 3-year, longitudinal study reported here, I explore the teaching and learn-

ing experiences of Laura and Elise (pseudonyms, as are all proper names throughout), 

two socially conscious novice bilingual teachers in a state where English-only poli-

cies were codified into law and with strict accountability measures. Both teachers at-

tended a post-baccalaureate teacher education program at a West Coast public uni-

versity and were purposefully selected (Patton, 2014) because of their expressed 

commitment to bilingual education and equity-oriented teaching. After completion of 

the credential program, both teachers were hired to work at the same dual language 

program (Spanish and English) in Valadez Elementary, an urban elementary school 

in which they also student taught during teacher preparation. Their shared commit-

ments and teaching context offer a unique opportunity to examine their teaching prac-

tices and how such practices are negotiated and grounded by their experiences and 

orientations developed over time. Studies that examine beginning teachers’ experi-

ences, especially with bilingual mathematics curricula, are scarce (Ingersoll & 

Strong, 2011; Sleeter et al., 2015) and important in advancing understanding of the 

processes in which teacher preparation programs can increase their potential to pre-

pare critical bilingual educators. Specifically, I explore two research questions: 

 

1. How do two novice bilingual elementary teachers organize mathematics 
learning for their emergent bilingual students?  

2. What opportunities arise for the teachers to learn to teach mathematics in bi-
lingual settings during teacher preparation and in the first 2 years of profes-

sional teaching?  

 

Data Sources 
 

In the study, I take a situated lens and view mathematics teaching, neither as 

static nor fixed, but rather as evolving as teachers learn to participate in and across 

communities (Wenger, 1998). As such, this study was longitudinal in nature, and eth-



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  112 

nographic methods (Patton, 2014) were used to collect data over a 3-year period en-

compassing 1 year in the teacher preparation program and the teachers’ first 2 years 

of professional teaching after graduation. To examine their teacher preparation, I at-

tended and took field notes at the two-quarter (10 weeks per quarter) elementary 

mathematics methods course. In addition to the field notes, all teaching artifacts from 

the mathematics methods course (e.g., course syllabi, PowerPoint slides, handouts, 

assignments) were gathered. Additional publically available programmatic data (e.g., 

program design, course sequence, syllabi of courses) were collected that may have 

emphasized pedagogies for working with emergent bilinguals: child development, 

educational equity, and theories and methods in English language development. 

As I sought to investigate their mathematics teaching after graduation, the 

main source of data gathered during the teachers’ first 2 years of professional teach-

ing were classroom observations. During the classroom visits, I observed their 50- 

to 75-minute mathematics lessons and followed each lesson observation with a 30- 

to 60-minute post-lesson interview, during which I interviewed the teachers about 

their mathematics lesson, their teaching, and the ways they saw themselves devel-

oping as teachers. All classroom lessons were videotaped and later transcribed and 

interviews were audiorecorded. Classroom artifacts (e.g., lesson plans, students’ 

assignments, class generated work), e-mail exchanges, and informal participant in-

terviews with teachers and other school personnel were gathered and served as sec-

ondary data sources. 

  

Data Analysis 
 

Data analyses occurred in two phases. The first phase examined how the 

teachers organized mathematics learning for their emergent bilingual students, 

drawing on prior literature and the three dimensions of principled instruction that 

framed the study: types of language supports, authority of knowledge, and reper-

toires of language practices. 

The software program Dedoose3 was used to review video data repeatedly for 

close attention to the details in classroom interactions. Dedoose allows for microa-

nalysis of video interactions and provides a time mark and key code deduction so 

that the data can be organized into themes. Data were coded according to the 

aforementioned three dimensions of focus. Each videotaped lesson was analyzed 

systematically, identifying aspects of the lesson and specific teaching practices re-

lated to types of language supports, authority of knowledge, and repertoires of lan-

guage practices (Strauss & Corbin, 1998). Data were also coded for the type of se-

miotic resources deployed by the teacher and student during whole classroom inter-

actions. Multimodal analysis (Jewitt, 2009; Kress, 2010) provides insight to the 

manner in which verbal language is enacted as well as the nonverbal in social inter-

                                                 
3 For details about Dedoose see http://www.dedoose.com. 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  113 

actions. This type of analysis is particularly helpful in research examining bilin-

gual/multilingual learners as students often rely upon semiotic resources other than 

spoken language to participate (Cummins, 2000; DaSilva Iddings, 2005; 

Domínguez, 2005; Turner et al., 2013). Appendix A provides the practices found in 

the data at this stage. 

Analysis was conducted on all videotaped classroom data. A research assis-

tant and I reviewed the data independently and met regularly to discuss reoccurring 

themes, coding categories, and interpretations. When divergent opinions existed, 

data were reviewed multiple times until an agreement was reached (Miles, Huber-

man, & Saldaña, 2014). Themes and interpretations were analyzed and compared 

across data sources: videotaped lessons, interview transcripts, and field notes 

(Fielding & Fielding, 1986). While the videotaped mathematics lessons allowed us 

to examine the classroom interactional architecture, the interviews and field notes 

provided a window into the intentions and reasons behind the teachers’ actions. 

Teachers’ socialization into the teaching profession begins as students in K–

12 classrooms, throughout teacher preparation, and continues as they encounter the 

reality of schools as they begin their professional lives (Feiman-Nemser, 1983; Lor-

tie, 1975; Zeichner & Gore, 2010). The goal of the second phase of analysis was to 

examine the professional opportunities that arose for the teachers during teacher 

preparation and in professional developing settings; however, their earlier K–12 

schooling experiences are included as these experiences also shape their percep-

tions of teaching. 

The second phase builds from earlier analysis of the interview data. The inter-

view data, field notes of observations at the school site and the mathematics meth-

ods courses, and programmatic data (e.g., teacher education program design, learn-

ing outcomes, course syllabi) were reviewed to examine learning opportunities that 

arose for the teachers during their K–12 setting, teacher preparation, and profes-

sional development settings related to bilingual mathematics teaching and learning. 

Analysis consisted of identifying the teaching repertoire privileged in the data 

sources. The notion of repertoire is taken from Bernstein (1996) and refers to the set 

of symbolic and material resources selected and configured to shape classroom 

practice. In this case, I examined the specific instructional materials, pedagogic re-

sources and strategies, and arrangement of task sequence privileged in the profes-

sional learning settings as identified in the interviews or in the programmatic data 

(Strauss & Corbin, 1998). Teacher portraits were written for each teacher highlight-

ing their learning experiences in K–12 schools, within teacher preparation, and after 

graduation (Miles et al., 2014). To verify and confirm interpretations of the data, I 

triangulated the various data sources (i.e., field notes, interview transcripts, field 

notes, and teaching artifacts) and teacher portraits were shared with each teacher to 

ensure accurate reflection of their experiences (Lincoln & Guba, 1986).  

 



 

 

 

 

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Findings 
 

In this section, I present the findings in two parts. The first part presents find-

ings from the study organized in three episodes to demonstrate the array of interac-

tional encounters the students navigated and the role of the teacher in shaping the 

classroom interactions at three timepoints during their second year of professional 

teaching. While their teaching repertoires were relatively consistent, both Elise and 

Laura felt more confident in their mathematics teaching by their second year. In the 

second part, teacher portraits that connect their teaching practices to their opportu-

nities to learn are shared. 

 

Organization of Mathematics Learning 
 

Both Elise and Laura utilized similar pedagogical strategies that leveraged 

students’ cultural and linguistic identities, but how these strategies were used re-

vealed differing ideologies about learning. These differing ideologies are discussed 

in relation to the three dimensions of principled instruction that guided the study: 

types of instructional supports, authority of knowledge, and repertoires of language 

practices. 

Types of instructional supports. Laura and Elise created a classroom envi-

ronment where talking about language was as important as the mathematics lesson 

itself. In their lessons, Laura and Elise structured instructional time for language to 

serve as the topic of classroom discussion. In their practices, they used various 

strategies from research. They encouraged students’ use of their native or hybrid 

language practices, modeled the target language in the lesson, and connected lan-

guage with mathematical representations (e.g., pictures, words, numbers, gestures; 

De Jong & Harper, 2008; Dutro & Moran, 2003; García et al., 2008; Moschkovich, 

2002, 2007, 2012; O’Halloran, 2005). 

While both teachers used similar strategies, there were variations in the way 

these strategies were enacted. An analysis of typical classroom interactions is illus-

trative. In Laura’s class, mathematics instruction followed a consistent routine. The 

class began with a math talk activity in which students explored mathematical con-

cepts or number patterns and relationships. The class then engaged in a problem-

solving activity with students working independently or in pairs. A public sharing 

of student solutions followed. 

Figure 1 and the discussion to follow are excerpted from the public sharing of 

a lesson taught in Laura’s second year. Here, her first-grade students were intro-

duced to a join change-unknown word problem (see Figure 1). 

 



 

 

 

 

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Nostros tenemos __ fotos en el altar. (Los estudiantes ponen algunas fotos más.) 

Ahora hay ___ fotos in el altar. ¿Cuántas fotos ponen los estudiantes?  

[We have ____ photos on the altar. (The students put more photos on the altar.) 

Now, there are ___ photos on the altar. How many photos did the students put on?] 

 

3/10                    13/20                    83/100 

 
 

Figure 1. Join change-unknown word problem. 

 

Excerpt 14 Laura [November, Year 2]: 

  

(Sammy walks up and stands next to Laura as she writes his name below the 

problem.) 
 

T: Sammy, EXPLICANOS esta idea. ¿Como contaste? / Sammy, EX-

PLAIN the idea to us. How did you count? 
 

S: Yo conté en mi mente (points to forehead) que 3 Y 1, 2, 3, 4, 5, 6, 7 

más y 3, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. (counts with fingers) / I counted in my 

mind 3 AND 1, 2, 3, 4, 5, 6, 7 plus 3, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 
  
T: ¿Oh Sammy, entonces empezaste con el número TRES? / Sammy, so you 

started with the number THREE?  
 

S: Si (Laura writes 3 below Sammy’s name.) / Yes  
 

T: ¿Por qué empezaste con el número TRES? / Why did you start with 

the number THREE? 
 

S: Porque tenemos TRES (points to the 3 written in the word problem dis-

played) y los estudiantes pone algunos mas (points to the problem again). / 

Because we have THREE and the students put some more. 
 

T: ¿Oh tenemos TRES fotos (finger circling the 3 in the word problem) en 

EL ALTAR y los estudiantes PONEN ALGUNAS MAS? / Oh we have 

THREE photos on THE ALTAR and the students PUT SOME MORE? 
 

                                                 
4 The following transcription symbols are used in all transcript excerpts / English translation follows  

( ) Description of nonverbal communication such as gestures, gaze, movement, and so forth 

Capital emphatic stress 

 Indicates rising tone 

T = teacher; S = student; SS = students 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  116 

T: Dígale a su amigo lo que recién dijo Sammy. / Tell your friend what 

Sammy just said. (Walks around listening to student partner talks.) 
 

T: ¿POR QUE Sammy empezó con el número TRES? ¿Quién nos puede 

decir? / WHY do you think Sammy started with the number THREE? Who 

can tell us? 
 

S1: Porque los otros tienen tres fotos y los estudiantes PONEN ALGUNOS 

MAS, don’t know how much (Lea points to “ponen algunas fotos más” in 

the word problem). Pero, al final es 10 FOTOS / Because we have three 

photos and the students PUT SOME MORE, don’t know how much more. 

But at the end, it’s 10 PHOTOS. 
 

T: ¿OH AL FINAL ES DIEZ? Okay, entonces voy a escribir diez (writes 10 

on the same line as the 3). Y sabemos que ahora hay diez fotos. ¿Cómo lle-

gaste de TRES a DIEZ? ¿Qué hiciste? / SO AT THE END THERE IS 

TEN? Okay. I’m going to write ten. We know that now there are ten photos. 

How did you get from THREE to TEN? What did you do? 

 

The vignette above documents the starting conversation around Sammy’s 

strategy. Figure 2 provides an image of Laura’s documentation of the student’s 

strategy on chart paper as the class examines his problem solving together. The 

whole class discussion around Sammy’s strategy lasted for over five minutes. 

Multiple students explained Sammy’s strategy and connected it to the problem’s 

context. By doing so, students were required to explain Sammy’s sense-making and 

the underlying characteristics of a change-unknown problem type. Exemplified here 

is Laura’s integrated approach to learning language and the math content. Students 

arrived at meanings and definitions as they engaged in analysis of each other’s 

reasoning and problem solving.  

 

 
 

Figure 2. Documentation of Sammy’s strategy. 

 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  117 

Elise’s classroom interactions followed a different pattern. Elise set a specific 

time before the math content instruction to teach language. Dutro and Moran (2003) 

term the teaching of vocabulary prior to content instruction “frontloading 

vocabulary.” The excerpt that follows captures the beginning of math instruction 

from a lesson observed during a similar time to Laura’s; this lesson occurred during 

Elise’s second year of teaching. Here, Elise’s second-grade students are learning to 

tell time: 

  

(Students are standing and holding one arm up. Elise is standing in front of 

the class with her right arm extended out.) 
 

T: VAMOS a aprender acerca de LAS HORAS del reloj. / We are going to 

learn about THE HOURS on the clock.  
 

SS: Vamos a aprender acerca de las horas del reloj. / We are going to learn 

about THE HOURS of the clock.  
 

T: Punto / point 
  
SS: Punto / point 
  
T: Quince / fifteen (left hand is extended straight out to represent the mi-

nute hand at the 15-minute mark on a clock) 
 

SS: Quince / fifteen (models the teacher with left hand extended straight 

out to represent the minute hand at the 15-minute mark on a clock) 
 

T: Treinta / thirty (left hand moves clockwise to the position of the 30-

minute mark on a clock) 
 

SS: Treinta / thirty (follows after the teacher with left hand moves clo-

ckwise to represent the 30-minute mark on a clock) 
 

T: Cuarenta-cinco / forty-five (left hand moves clockwise to the position of 

the 45-minute mark on a clock) 
 

SS: Cuarenta-cinco / forty-five (left hand moves clockwise to the 45-minute 

mark on a clock)  

 

Elise organized instruction into two parts. Instruction began with students en-

gaging in a vocabulary activity using total physical response—in this case, using 

their hands to represent the minute hand and identifying the parts of an analog 

clock. The clock vocabulary activity lasted for 10 minutes in which students en-

gaged with mathematical terms across representations (e.g., gestural, visual, graph-

ical, oral). After the song, Elise and the students read from the vocabulary word 

bank posted on the wall with students echoing back in choral response as Elise 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  118 

pointed to and read the written math vocabulary word or phrase. In other activities, 

students created a human number line (choral response of double digit numbers), 

searched for geometric shapes in the classroom (identification of geometric struc-

tures), and recited the place value chart (naming the value based on its place, or po-

sition, on the chart). Elise’s goal for her vocabulary activity was to familiarize stu-

dents with the vocabulary for the lesson. This format of front-loading the vocabu-

lary followed by engagement with the mathematical content occurred in the majori-

ty of the math lessons observed in her class. 

Authority of knowledge. Students in all classrooms traverse an array of inter-

actional practices where power and status impacts students’ access to learn and to 

demonstrate their learning (Cazden, 2001; DaSilva Iddings, 2005). Analysis of 

classroom interactional patterns provides insight to issues of agency and positioning 

in the classroom (Hand, 2012; Jewitt, 2009; Kress, 2010). Another difference iden-

tified during analysis was the positioning of competence in the two classrooms. 

Laura attended closely to the nature of the classroom culture and tried to cen-

ter instruction on inviting students to become a part of the classroom community. In 

the episode shown earlier, Laura positioned herself as the facilitator as well as a 

learner within the classroom community. As shown in the vignette above, Laura 

often adopted a posture of uncertainty (“¿POR QUE Sammy empezó con el núme-

ro TRES? ¿Quién nos puede decir? / WHY do you think Sammy started with the 

number THREE? Who can tell us?”) to open up space for students to take on the 

expert role (“Dígale a su amigo lo que dijo Sammy ahora. / Tell your friend 
what Sammy just said.”). 

In a later component of the public solution share, Laura called on a third stu-

dent, Jesus, to explain his strategy. Jesus was a quiet student, who had never spoken 

out during class discussions on previous visits. What follows is the whole class dis-

cussion after Jesus shares his strategy with the class: 

 

T: EXPLICALE a su amigo lo que hizo Jesus. / EXPLAIN to your friend 

what Jesus did. (The students talk amongst themselves, then Laura claps them 

to attention.) 
 

T: Okay. Alguien puede explicar porque Jesus EMPEZO con veinte y 

terminó con trece? Por que hizo ESTO? Jesus, escoge a otra persona que 

pueda explicar lo que hiciste. / Who can explain why Jesus STARTED with 

20 and ended with 13? Why did he do THIS? Jesus, pick another person 

who can explain what you did. 
 

(Jesus points to a student sitting near the back, who comes and stands next to 

Jesus.) 
 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  119 

S3: Habían trece al principio (uses finger to circle the thirteen crossed out 

dots that represent a visual of Jesus’s strategy shown on chart paper) y 

LUEGO habían veinte y ESTOS son los que ponen. / There were thirteen 

in the beginning and THEN there were twenty and THOSE are the ones 

they put there. 
 

T: Hmm, okay. / (puts finger on cheek pensively) Alguien. / Hmm, okay. An-

yone…  
 

S4: Porque Jesus did it BACKWARDS a Sammy… so it’s twenty take away 

thirteen is seven. / But Jesus did it BACKWARDS to Sammy… so it’s twenty 

take away thirteen is seven. 
 

T: Veinte quita trece es igual a siete y tú dijiste él lo hizo al revés a Sammy. 

Qué quieres decir? (writes 20–13=7 under the dot representation) / Twenty 

minus thirteen is equal to seven. You said that he did the reverse to Sammy. 

What did you mean? 
  
S5: Sammy counted up. Sustracción es opposite. / Sammy counted up. 

Subtraction is opposite. 

  

In the scenario above, Laura demonstrated some of the instructional moves 

used to distribute the knowledge authority between herself and among the students. 

One way is her consistent questioning, displaying a sense of wondering (“Alguien 

puede explicar porque Jesus empezó con veinte y terminó con trece? Por que hizo 

esto?” / Who can explain why Jesus started with 20 and ended with 13? Why did he 

do this?”). Laura regularly asked the class to explain the reasoning behind a 

particular strategy. This practice was a common move used by Laura to open the 

floor to peer mediation, putting the onus of explanation and analysis on the 

students. Students’ share-outs were not always fully articulated ideas, partial 

explanations were common, and students built upon each other’s explanations. 

During lesson reflections, Laura regularly discussed the intentional selection and 

sequencing of specific solution strategies as well as strategies to reconfigure the 

classroom social order for more meaningful and respectful dialogue. 

In another example, I highlight a lesson that took place during a similar period 

to illustrate differences between the two classrooms. During March of Elise’s 

second year, the class was on day two of working on regrouping in subtraction. The 

day prior, the class used concrete models to represent the process of regrouping: 

working with place-value blocks and place value charts to connect the procedural 

skill of regrouping (the decomposing of a 10 unit for 10 ones) to its conceptual 

basis (each place value position is related to the next by a constant multiplier of 10). 

Each student had on their desks a set of base-ten blocks and the place-value chart. 

The following dialogue is from the start of the lesson during which Elise 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  120 

demonstrated the process of regrouping from concrete to representational (see 

Figure 3)—connecting the written recording of each step of the procedure to the 

modeling with the base-ten blocks: 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

Figure 3. Teacher modeling of the problem on the overhead. 

 

T: Con los bloques de diez, les voy a dar un cuento sobre algo, así que tienen 

que escuchar calladitos. ¿Por ejemplo, si digo Diego tiene TREINTA-

CUATRO pájaros (Elise places the three base ten rods in the tens place and 

four units in the ones place of the place value chart, Figure 3), cuantas DE-

CENAS voy a poner? / With the blocks of tens, I’m going to give you a sto-

ry. So, you all must listen quietly. For example, if I say Diego has THIRTY-

FOUR birds, how many TENS am I going to have? 
 

SS: Tres (choral response) / Three.  
 

T: ¿Cuantas me faltan para ser TREINTA-CUATRO? (Elise points to the 

four ones units on the ones side of the place value chart.) / How many more 

do I need to make THIRTY-FOUR? 
 

SS: Cuatro (choral response) / Four  
 

T: (A few students have the base ten blocks in hand.) No es necesario tocar 

nada en este momento. (Elise looks around the room and waits for students to 

put down the base ten blocks.) Pero su hermanito, Allen, le abrió la jaula, y se 

le fueron VEINTICINCO. Se fueron VEINTICINCO pájaros ¿Que voy ha-

cer? Se fueron VEINTICINCO pájaros. ¿Qué vamos hacer? / You don’t 

need to touch anything right now. But his little brother, Allen, opened the ca-

ge, and TWENTY-FIVE left. TWENTY-FIVE birds left. What are we going 

to do? 

  



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  121 

SS: Vamos a quitar UNA DECENA. (choral response) / We are going to take 

away A TEN.  
 

T: No es necesario tocar NADA en este momento. (Elise waits for students to 

put down the base ten blocks in their hands.) / You don’t need to touch 

ANYTHING right now. 
 

T: ¿Si, vamos a quitar una decena y que se va hacer? / Yes, we are going to 

take away a ten and what is that going to make?  
 

SS: Pone unas unidades. / Put ones. (Elise takes away a ten rod from the ten 

place-value column.) 
 

T: ¿Las pongo en unidades? (Elise places the ten rod into the ones place va-

lue column.) / I put them with the ones?  
 

SS: No (choral response) 
 

T: ¿las voy a poner aquí? ¿Qué va a pasar? (Elise lifts the placed ten rod 

from the ones place-value column and holds it up for students to see.) / Am I 

going to put them here? What is going to happen? 
  
S1: Vas a quitar dos decenas. / You are going to remove two tens.  
 

T: ¿DOS decenas? (Elise holds up the ten rod in hand and shakes it.) / TWO 

tens?  
 

S1: No  
 

SS: Una decena. / One ten. 
   

T: Una decena. (Elise holds up the ten rod in hand and shakes it.) ¿Dónde se 

va a ir esta decena? / One ten. Where is this ten going to go? 
  
SS: En las unidades. / In the ones. (Elise places ten individual units in the 

ones column.) 
 

T: ¿Y vas a cambiarlas PORQUE? / And WHY are you going to change 

them? 
 

SS: por unidades, se fueron / for ones, they left  
 

T: ¿Veinticinco se fueron entonces ya puedo quitar veinticinco? / Twenty-five 

left and now can I take twenty-five?  
 

SS: Si (choral response) / Yes  

 

This vignette illustrates Elise’s developmental approach to linking a proce-

dural skill to a conceptual base. Elise was building on the concrete experience of 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  122 

the day prior addressing the representational stage—linking each step of regrouping 

to the written recording. She represented the minuend with the place value blocks 

and made trades (i.e., exchanging 1 for 10 in the position to the right) before sub-

tracting. This process of “trading all” before subtracting has been shown to help 

prevent errors in subtraction (Van de Walle, Karp, & Bay-Williams, 2010) and 

demonstrated Elise’s attention to students’ mathematical development. 

Elise regularly used physical and visual models to connect procedures and 

concepts; however, she also determined which models were used as well as the pro-

cess for problem solving. At one point, a student appeared to suggest a different 

way to subtract (“Vas a quitar dos decenas. / You remove two tens.”). It’s possible 

that the student may have wanted to subtract by place value—taking the two tens of 

25 from 34 first. Elise, however, redirected the students’ attention back to the tens 

rod in hand as she wanted students to regroup before subtracting. Elise, unfortu-

nately, often started lessons modeling a solution path for students. By providing a 

solution path, Elise unintentionally reinforced a hierarchy and dichotomy between 

the teacher and student and knower and learner. 

Repertoire of language practices. The last theme in the examples is related to 

the previous and documents the repertoire of language practices utilized. Both 

teachers created lessons that encouraged multi-modal and hybrid language practic-

es; however, there were variances in how the students’ language practices were lev-

eraged during instruction. 

The following scenario illustrates how Laura built upon students’ language 

practices over time. I discuss a visit made 2 months after the photo-problem lesson 

described in the earlier vignettes. The class regularly engaged in student-strategy 

share. In prior lessons, solution strategies were labeled with the students’ names 

(e.g., Rosa’s strategy). In this visit, the lesson focus was “estrategias para juntar / 

strategies for adding” to link each student’s addition strategy to its mathematical 

term. However, the definition was not given by Laura but unpacked by the students 

as they analyzed the strategy’s features: 

 

T: Cada vez que hacemos UN PROBLEMA hay personas que comparten 

su IDEA, verdad? / Every time we do A PROBLEM, there are other peo-

ple who share the same IDEA as you, right? 
  

SS: Si / Yes  
 

T: Si, cada vez que lo hacemos. Pero estaba pensando que sus ideas no tie-

nen nombre. Entonces si yo le digo, as la idea de Caleb, NO SE de qué es-

toy hablando. Entonces HOY quería hacer UNA LISTA de las ideas que 

tenemos. Entonces llamaré esta hoja estrategias para juntar. ¿Que es una es-

trategia? ¿Quien ha escuchado esta palabra? (Laura points to the words 

“Estrategias para juntar/strategies for adding” on the chart paper. Another 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  123 

sheet of chart paper covers this one. Laura lowers the top paper to reveal the 

first strategy “Modelar de uno en uno (Modeling one by one).” The strategy 

title and the visual example are shown.) / Yes, whenever we do it. But what 

I was thinking that your ideas do not have a name. Therefore, if I tell you 

all to do Caleb’s idea, I DON’T KNOW what I am talking about. TODAY, 

I would like to do A LIST of the ideas that we have. Therefore, I call this 

page strategies for adding. What is a strategy? Who has heard this word? 
  

T: La primera estrategia es de Ella. Su estraegia el nombre de su estrategia 

es modelar de UNO EN UNO. Esta es una manera que MUCHOS de uste-

des están USANDO modelar de uno en uno. Y que están observando de la 

estrategia de Ella? Que ha hecho. Hmmm. (Laura puts her hand to her face 

pensively. Many students in the class follow the same pose.) / The first strate-

gy is Ella’s. Her strategy’s name is modeling ONE BY ONE. This is one 

strategy that MANY of you USE. What do you observe about Ella’s strate-

gy? What has she done? Hmmm. 
 

T: TODOS piensa su estrategia. ¿Que están observando? ¿Que están obser-

vando de la idea de Ella, Alex? / EVERYONE think about her strategy. What 

do you observe? What do you observe about Ella’s idea, Alex? 
 

S1: (Alex walks up to the poster and points at Ella’s strategy) um, tiene 

quince y catorce-trece the same one as this (Alex walks over to the class math 

wall) because quince y trece and then right there (points to strategy on a 

poster on the math wall where the strategy shown is also a modeling all strat-

egy) he put quince y trece mas, but then she counts each one – one, two, 

three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, 

fifteen, (touches each drawn unit and counts out loud) and then one, two, 

three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, 

fifteen, and then you also could do this. / um, there’s 15, and 14, 13, the 

same one as this because 15 y 13 and then right there (points to strategy on 

another poster where the student is also modeling all) he put 15 and 13 more, 

but then she counts each one, two, three, four, five, six, seven, eight, nine, ten, 

eleven, twelve, thirteen, fourteen, fifteen, and then one, two, three, four, 

five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen. 
 

T: Oh, Entonces estas diciendo, el está diciendo que en la idea de Ella cuen-

ta TODAS LAS COSAS. Entonces Alex dijo que cuenta cada palo, ¿ver-

dad? Cuenta de uno en uno, voy a escribir eso. (She writes “cuenta todas 

las cosas / counts one by one” under the modeling one by one representa-

tion.) / Okay, then what you are saying, he is saying that Ella’s idea is to 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  124 

COUNT EVERY THING. Then, Alex said to count each of the lines, 

right? Count one by one, I am going to write this. 
  

T: Algo más que observan de su idea de Ella? Ana? / Is there anything 

else you observe about Ella’s idea? Ana?  
 

S2: Una manera fácil también is to take quince FIRST and add uno, dos, 

tres, cuatro, cinco, seis, siete, ocho, nueve los demás más los demás le va 

hacer fácil solo hacer números. / An easy way to also do this is to FIRST 

take the first fifteen plus the other one, two, three, four, five, six, seven, eight, 

nine and then the rest of them plus the rest of them. It will be easier to just 

use those numbers.  
 

T: Estas diciendo que sería más fácil usar números? / You are saying that it 

would easier to use numbers? 
  

S2: Si porque te vas a hacer CANSADA de escribir muchas muchas lí-

neas. / Yes because you will get TIRED of drawing so many, many li-

nes.  
 

T: En esa tienes que escribir MUCHAS cosas. Voy a escribir eso también, 

(Laura writes “escribes muchas cosas / writes down a lot” on the chart pa-

per.) Addy algo más? / In this one, you must write MANY things. I am 

going to write this one too. Addy, anything else? 

 

The interaction above highlights some of the ways Laura drew on students’ 

language practices. First, she used students’ ways of thinking and reasoning about 

the strategies as the focus of discussion. Laura’s classroom walls were filled with 

posters of student strategies and their representations (e.g., drawings, words, equa-

tions) gathered from lessons past. As in the vignette above, these posters were often 

referenced by Laura and the students during instruction. 

The students also moved in and out of a range of language practices (Spanish, 

English, and code switching between Spanish and English). This flow of language 

practices was most observed when students were trying to communicate their rea-

soning and sense-making. It appeared Laura’s students often articulated the aca-

demic language of mathematical concepts and numbers using the school-based dis-

course heard in class and would use their everyday, native language (Spanish or 

English) to provide illustrations (“una manera fácil también is to take quince first 

and add uno, dos, tres, cuatro, cinco, seis, siete, ocho, nueve los demás más los de-

más le va hacer fácil solo hacer números”). 

Notice in the vignette above, Laura accepted Alex’s response but rephrased a 

more complete response in Spanish so the students could hear the formal Spanish 

discourse (“El está diciendo que en la idea de Ela cuenta todas las cosas. Entonces 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  125 

Alex dijo que cuenta cada palo, ¿verdad? Cuenta de uno en uno voy a escribir eso. 

/ Okay, what you are saying, he is saying that Ella’s idea is to count. Alex said to 

count each of the lines, right? Count one by one, I am going to write this.) 

Semiotic resources other than spoken or written language, including gestures 

and physical movement, were often used. In the example above, Alex pointed to 

another student example of a modeling one-to-one strategy to explain how the strat-

egy required students to “count one by one.” By revoicing his explanation in Span-

ish using the mathematical terms and writing down his observation on the poster 

sheet, Laura showed that all forms of communication—verbal or nonverbal, Eng-

lish or Spanish—were valued. 

In Elise’s classroom, students also moved between language practices in their 

student-to-student interactions and during recess; however, hybrid and multimodal 

practices were seen less frequently than in Laura’s case during classroom discus-

sions. I contrast here Laura’s use of students’ language practices during strategy 

shares with a vignette of Elise’s class discussion of a strategy for subtraction with 

regrouping. The strategy is not the traditional U.S. subtraction wherein the algo-

rithm subtracts from right to left, one place-value column at a time, regrouping as 

necessary. In this strategy, the students are subtracting numbers in expanded form, 

the subtrahend and minuend are written in expanded form in vertical order with the 

same place value arranged in columns and differences found for each place value:  

 

T: (Points to the number 427 written in the problem 427–182.) ¿Okay, que 

son las escenas de 427?  / Ok, what is the expanded form for 427? 
 

S1: Cuatro / Four  
 

T: ¿Cuatro, y como vamos hacerlo? / Four, how are we going to do it?  
 

S2: Cuatrocientos / 400 
  
T: Cuatrocientos / 400 (Writes down 400, 20, 7 on the worksheet projected on 

the overhead, then points to the 20 and 7.) y que es ESTO? veinte y siete y 

ciento ochenta y dos (points to the 182) / And what is THIS? 20 and 7. And 

182.  
  
SS: Cien (writes down 100 below the 400 in the expanded form of 427) / 100  
 

SS: 80 and 2 (writes down 100, 80, 2 in columns by place value below the 

400, 20, and 7) 
 

T: Ochenta y dos. ¿Qué ES siete menos dos? / Eighty and two. What IS sev-

en minus two? 
 

SS: Cinco (writes 5 down and points to “20-80”) / Five  
 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  126 

S3: Sesenta / Sixty 
  
T: Espérese ¿Cómo se puede restar 20 de 80? ¿Qué le vas a decir? / Hold 

on, how can you subtract twenty from eighty? What are you going to say? 
 

S4: Puedes prestar de los 400. / You can borrow from the 400.  
 

T: ¿Le PRESTAMOS 400 y ese se convierte¿ (points at the 400) / We 

BORROW from 400 and what does this one convert into? 
  
S4: Este se convierte en 300. (Elise crosses out the 400 and writes 300 

above.) / That one turns into 300.  
 

T: ¿Por qué? / Why? 
 

S4: Porque le presaste uno prestado uno al 20. / Because you lend one to the 

20.  
 

T: ¿Por qué le prestas? / Why do you lend it? 
  
S5: Porque 80 es más que 20. / Because 80 is more than 20.  

 

In the interaction above, the class used verbal and nonverbal communication 

(gestures and the expanded form) to make sense of the regrouping process in sub-

traction. What follows is a brief episode of the discussion for regrouping in the next 

problem: 

 

(The class already decomposed 639 and 256 into place value in expanded 

form together, and this is shown on the paper on the overhead.) 
 

T: ¿Entonces que estamos haciendo con nueve menos seis? / So what are 

we doing with 9-6? 
 

S1: Tres / Three  
 

T: ¿Jay, que vamos hacer con treinta y cincuenta? / Jay, what are we do-

ing with thirty and fifty? 
 

S6: Puedes PRESTAR de seiscientos y poner quinientos. / You BORROW 

from six hundred and you have five hundred. 
 

T: Quinientos? (Elise crosses out the 600 and writes 500 above) / Five hun-

dred?  
 

S6: Quinientos / Five hundred 
 

T: ¿Pero porque le vas a dar una centena al treinta? / But why are you 

giving one hundred to the thirty? 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  127 

 

S6: Porque cincuenta es MAST que treinta / Because 50 is MORE than 

30.  

 

The vignettes provide a window into the classroom discourse around regroup-

ing for two problems; however, the same discursive pattern continued for the re-

maining 30 minutes of the lesson. As seen above, little language hybridity occurred 

during the strategy shares. The class discussion was only in Spanish. This use can 

be expected, as students in Elise’s classroom were a grade above Laura’s and had 

more opportunities to develop their Spanish proficiency. However, there was also 

less variance in what was said. Note how student responses for both problems for 

the regrouping process (“Porque le prestaste uno al 20.” / “Because you lend one 

to the 20.”; “Puedes prestar de seiscientos.” / “You borrow from 600.”), and the 

reason why regrouping was needed (“Porque 80 es más que 20.” / “80 is more 

than 20.”; “Porque cincuenta es más que treinta” / Because 50 is more than 30.) 

were very similar. In Elise’s class, Spanish, mathematical representations and ges-

tures were frequently used, but she modeled the representations and gestures first. 

Student responses often consisted of a repetition of her words and phrases (“you 

borrow one”; “more than”). 

  

Examining Teachers’ Opportunities to Learn 
 

It is important to connect Laura’s and Elise’s teaching practices to their 

opportunities to learn to teach. What follows are teacher portraits for each teacher’s 

K–12 schooling, motivation for bilingual teaching, and their professional learning 

opportunities during teacher preparation and after graduation. 

Laura. Laura grew up in a bilingual household with a stepmother who only 

spoke Spanish at home, while her father only spoke in English. Laura took Spanish 

in middle and high school, and majored in Spanish in college. After college 

graduation, Laura did not plan to pursue a career in teaching but struggled for a year 

to find work. With both parents as teachers, Laura saw teaching, particularly a 

career as a bilingual teacher, as a stable career option.  

Laura attended a teacher preparation program located in a city with a growing 

emergent bilingual student population. The program is designed to prepare teachers 

for the complexities faced in their teaching placements. The credential program 

focuses on developing pre-service teachers’ competencies in four areas: (a) 

developing an inquiry stance; (b) supporting second-language learners; (c) 

collaborating with faculty, peers, and mentors to continually improve practice; and 

(d) appreciating unique resources students bring to the classroom. 

During teacher preparation, Laura took courses on language acquisition, 

multicultural education, child development, and mathematics methods. Course 

readings and assignments focused on strengths-based approaches to learning, but 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  128 

the context of teaching language learners explored theories and teaching practices 

for English language acquisition only. Theoretical underpinnings, practical 

applications of bilingual education, and critique of English hegemony in U.S. 

schools were not included in the syllabi reviewed. 

As a teacher candidate working toward a bilingual authorization, Laura was 

one of four students in a cohort of over 60 preparing to teach in bilingual settings. 

During the last interview of each year of the study, Laura was asked to reflect upon 

the extent to which she relied on materials learned from her university courses. 

Laura stated that she relied very little on her teacher preparation experiences. Her 

coursework felt unrelated to “the realities of teaching Spanish and English to 

students” and learning to teach emergant bilingual students for her was rooted in 

“professional developments outside of and after the credential program.” 

Laura student-taught in a Spanish/English bilingual kindergarten classroom at 

Valadez Elementary School when the bilingual education program was in its 

second year. At the time of student teaching, bilingual classes were only offered in 

kindergarten and first grade and only a small fraction of the school staff were 

bilingual. The student demographics were also linguistically diverse. The school 

had a predominantly Latin@ student population (over 85%); half of the children in 

school were native Spanish-speakers and the other half native English-speakers. 

Instruction at her grade level was a 90/10 model, with curriculum instructed 90% in 

Spanish and 10% in English. She described her student-teaching experience as vital 

in her development as a bilingual teacher: 

 
I spoke Spanish, but the academic language needed to teach English, mathematics, and 

science required words I never used before. I never had to speak Spanish all day; my 

own Spanish improved so much during that time. Then having to consider students’ 

Spanish and English language development while planning lessons every week…  

 

Laura attributed the learning of academic vocabulary, analysis of the basic el-

ements of Spanish and English language structures, and instructional strategies for 

dual-language development to being situated at a field site where both Spanish and 

English were used for instruction. While the school offered a strong bilingual litera-

cy program, Laura noticed that the mathematics instruction at Valadez consisted of 

“just numbers and procedures” and varied greatly from the inquiry-based instruc-

tion of the mathematics methods course. During the summer after graduation, Laura 

asked her university field supervisor, Julia, for additional professional development 

opportunities and attended a math professional development course and made site 

visits to observe Cognitively Guided Instruction.5 

                                                 
5 Cognitively Guided Instruction, often abbreviated as CGI, is an approach to mathematics teaching 

that builds on children’s problem-solving strategies (see Carpenter, Fennema, Franke, Levi, & 

Empson, 1999). 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  129 

At the time of Laura’s hire, the state had just adopted new state standards and 

the principal at Valadez requested additional professional training for the staff. A 

district mathematics coach, who happened to be Julia, Laura’s university field su-

pervisor, came weekly for a year to work with the Valadez teachers. Laura worked 

with Julia biweekly for the entire year. By the third year of the study, Laura had 

developed a reputation for her strength in mathematics teaching and was selected to 

develop mathematics curricula for the district. Teachers from other schools and dis-

tricts interested in bilingual mathematics instruction responsive to students’ think-

ing frequently observed Laura’s classroom. 

Elise. Elise wanted to be a teacher since kindergarten and was in a dual-

language program herself until California’s passage of Proposition 227. The pas-

sage of the proposition led to the elimination of bilingual classes and Elise’s transi-

tion to English-only settings in second grade. Elise noted that she entered teaching 

with the intent to share critical knowledge with her Latin@ students. Deeply com-

mitted to supporting students’ self-identity, Elise was actively involved in a com-

munity-mentoring program in which she received mentoring as a teenager and sub-

sequently mentored Latina youths.  

Elise attended the same teacher preparation program as Laura but a year later. 

Similar to Laura, Elise was one of six bilingual teacher candidates in a cohort of 

over 60 preparing to teach in bilingual settings. When asked to reflect upon her 

teacher preparation, Elise talked about the utility of the course readings from the 

university math and literacy methods courses and how these resources were still 

referenced occasionally, but she also described her loneliness and marginalization 

in the program:  

 
It’s hard. All the materials I have in the classroom (now) are translations of English 

textbooks to Spanish. The Spanish is off and the stories don’t connect to my kids. It 

(was) the same in the program. Everything was about preparing us to work with stu-

dents in English. How (a)bout for those of us that are not White and teach in English 

and Spanish? 

 

Elise noted that the prospective teachers and professoriate in the teacher edu-

cation program were predominantly White and monolingual (in English); only one 

of her course instructor was bilingual and a person of color. Both Elise and Laura 

had the same instructors for multicultural education, English language acquisition, 

child development, and the mathematics methods courses.  

Preparation for teaching language was limited to one course devoted to “Eng-

lish language acquisition” in which she learned about specifically designed academ-

ic instruction for English (SDAIE) strategies. These language methodologies ex-

plicitly target vocabulary development through teachers’ active modeling and 

teaching of conversational and academic language through the use of visuals, 

demonstrations, and hands-on learning (Dutro & Moran, 2003; Krashen, 1989). 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  130 

Elise’s frontloading of language in which she emphasized key vocabulary and made 

the learning experience multi-sensory is representative. 

Elise student taught in a second-grade classroom at Valadez, where the dual 

language program was in its third year. With her master teacher, Claudia, Elise 

taught the first Spanish immersion second-grade classroom at Valadez. Instruction 

in second grade was an 80/20 model, in which Spanish was used 80% and English 

20%. Mathematics was taught in Spanish, and English as a second language class 

was offered to students in the afternoon. 

After graduation, Elise was hired to teach in the same grade, and Claudia be-

came her grade-level colleague. Claudia and Elise were the only bilingual second 

grade teachers at the school site, and Valadez was the only school in the district to 

offer a dual language English and Spanish immersion program. Elise worked close-

ly with Claudia throughout the 3 years of the study. Elise discussed curricular deci-

sions with Claudia daily and stated that she “basically did the same things” as 

Claudia. Claudia regularly shared lessons and student worksheets with her. Given 

that there were limited instructional materials available in Spanish, Elise found the 

resources extremely helpful. 

During the 3 years of the study, Elise attended two professional development 

workshops on CGI offered by the district and made three site visits to observe CGI 

at a local school. The school also received a 1-week workshop on their new math 

textbook during Elise’s second year of professional teaching. Elise regularly attend-

ed professional development workshops centered on discussion of language and 

literacy. However, none of these workshops combined discussions of language with 

mathematics subject-matter knowledge. 

Elise talked about the school site as a place of struggle. The dominant pres-

ence of English diminished the place of Spanish in the school-wide context. Span-

ish was rarely heard on the playground or in the lunchroom. Even in the classroom, 

students often spoke in English during group work. Elise felt a responsibility to ori-

ent students to the importance of Spanish. An avid proponent of students’ linguistic 

development, Elise intentionally structured time at the start of math instruction to 

work on students’ disciplinary literacies in Spanish.  

 
Discussion 

 

The first part of the findings examined Laura and Elise’s mathematics teach-

ing, and the second examined the teachers’ opportunities to learn. Study findings 

confirm the importance of cultivating a linguistically diverse teaching force noted 

in previous research (e.g., Achinstein & Ogawa, 2011; Sleeter et al., 2015; Villegas 

& Irvine, 2010). Both Laura and Elise valued and honored students’ linguistic, cul-

tural, and experiential knowledge. Their mathematics lessons directly attended to 

the role of language in mathematics learning, promoted multimodal communica-



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  131 

tion, and encouraged students’ use of their native language. While both teachers 

used similar instructional strategies, how the strategies were used revealed differing 

orientations toward learning: learning as acquisition and learning as participation. 

A widely employed conceptual lens for understanding learning is the acquisi-

tion model, which views learning as the acquisition of specific knowledge and con-

cepts. Both teachers led lessons with the goal of supporting students to “acquire” 

knowledge of specific concepts (e.g., part-whole relations) and skills (e.g., subtrac-

tion regrouping). Within this model, new knowledge is acquired and accumulates 

through a developmental progression, building from one concept to the next, 

through teacher facilitation and scaffolding. Conceiving of learning as a develop-

mental trajectory provides a useful model for building students’ Cognitive Academ-

ic Language Proficiency (CALP), or formal academic language, from Basic Inter-

personal Communication Skills (BICS), or day-to-day language (Cummins, 1984). 

The acquisition model encourages a trajectory from concrete, representational, to 

abstract stages for mathematical concepts. Research on CGI demonstrates the bene-

fits of teachers’ understanding of learning trajectories and pedagogical considera-

tions responsive to students’ development (see, e.g., Carpenter et al., 1999). 

As Sfard (1989) has argued, however, an overemphasis on one metaphor for 

learning can be dangerous. Conceiving learning exclusively within the acquisition 

model can lead to treatment of concepts as objects of learning and separate from the 

context in which they are learnt. A widely endorsed teaching practice is the front-

loading of vocabulary, demonstrated by Elise, in which language is taught inde-

pendently of its application. The idea of concepts building upon each other has also 

led to viewing development as linear, to notions that certain vocabulary or mathe-

matics concepts must be taught before students’ engagement in rich mathematical 

talk or more challenging tasks, and to the idea that knowledge must transfer from 

teacher to students. This linear transfer metaphor is widespread within Western ed-

ucation, shapes curriculum and pedagogy, and shaped Elise’s mathematics instruc-

tion. This linearity can be seen in how Elise followed a lesson structure in which 

she gradually decreased the amount of “scaffolding” provided through repetition 

until students were able to replicate the learnt action on their own. 

Just as dangerous is to view learning only as participation. While the acquisi-

tion model is entrenched in what happens in individual minds, the participation 

model views learning as situational, indexically bound to social context (e.g., 

Brown, Collins, & Duguid, 1989; Lave & Wenger, 1991). The participation model 

can embed learning so completely within a given context that the individual is lost 

(Sfard, 1998). 

Therefore, a balanced model of learning is needed that considers both acquisi-

tion and participation in which learning emerges through interaction between the 

individual and the environment (Engestrom, 2001). Laura’s classroom reflects this 

balanced approach. Laura strategically designed mathematical tasks and selected 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  132 

student strategies to support the acquisition of specific concepts and skills based on 

her knowledge of students’ development. Classroom interactions supported indi-

vidual and collective understandings as Laura and her students embraced and built 

upon the repertoire of practices available. By valuing diverse voices and perspec-

tives, Laura created an egalitarian space for learning in which students saw them-

selves in what was being taught. Students who might have been dismissed as less 

capable in more traditional settings (e.g., developing in school-based discourse, 

solving problems using direct modeling, having only partial answers) were recog-

nized by others and saw themselves as integral to the learning community. 

While both models, learning as participation and learning as acquisition, are 

recognized in research and practice, the acquisition model is dominant. School-

based learning often follows the assumption that students should receive and learn 

what is clearly communicated and explicitly taught by the teacher (Freire, 

1970/2000; Valenzuela, 2002). A balanced model reminds us that students’ learning 

as well as the teachers’ learning are bound by the elements available in the ecologi-

cal system. This boundness brings to light the significance of Elise and Laura’s own 

learning to teach in bilingual settings. 

The study findings, like those by Achinstein and Ogawa (2011), challenge the 

cultural match assumption. As Achinstein and Ogawa argue, “just being a person of 

color does not guarantee effectiveness in teaching students of color, nor do new 

teachers of color necessarily have knowledge of pedagogy for diverse students” (p. 

15). The cultural match assumption downplays the significance of teacher prepara-

tion and of the cultural, structural, and institutional racism that shape K–12 school-

ing. 

With few exceptions, most states do not require teacher preparation programs 

to offer coursework specific to teaching emergent bilinguals (López et al., 2013), 

and most teacher education programs require only one “diversity” course (Aquino-

Sterling et al., 2016; Sleeter et al., 2015). The teacher preparation program attended 

by Elise and Laura provided more: a program mission for high-quality teaching in 

culturally and linguistically diverse communities, two mathematics methods cours-

es, and a few “diversity” courses. Still, their preparation was inadequate to class-

room and programmatic needs. 

Analysis of the program revealed that their teacher preparation was not de-

signed to build upon the strengths and needs of bilingual teachers. The two-

semester mathematics methods course focused on developing high-leverage instruc-

tional practices to support students’ development of central mathematical concepts; 

however, there was little emphasis on preparing teachers to support students’ de-

velopment of language skills to engage meaningfully in disciplinary discourse. As 

well, there was an underlying assumption that these teaching practices were the 

“right” teaching methods and strategies for all students and all teachers. This one-

size-fits-all model ignores the Western, assimilationist perspective that dominates 



 

 

 

 

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Journal of Urban Mathematics Education Vol. 10, No. 2  133 

U.S. schools (Gutiérrez, 2002; Khisty & Willey, 2008; Ovando, 2000; Sleeter, 

2001) and fails to leverage the different assets and needs bilingual prospective 

teachers bring to teacher preparation (Arce, 2004; Gomez et al., 2008; Musanti et 

al., 2009; Sleeter et al., 2015). 

Language is an important medium through which equity and inequities are 

structured and sustained in K–12 classrooms and in teacher education (Aguino-

Sterling et al., 2016; Gomez et al., 2008; Khisty & Willey, 2008; Sleeter et al., 

2015). The study findings, as well as others (e.g., Moschkovich, 2012; Remillard & 

Cahnmann, 2005; Setati, 2008; White et al., 2016), create an imperative for the de-

velopment of linguistically and culturally relevant mathematics teacher education.  

This study is premised on the idea that bilingual teachers are “learners and in-

novators with cultural and linguistic repositories ... to challenge and inspire students 

toward academic achievement” (Gist, Flores, & Claeys, 2015, p. 29). However, the 

study findings serve as a reminder that teacher educators (and researchers) must be 

careful not to romanticize bilingual teachers nor presume that bilingual fluency or 

ethnic identity alone are sufficient to challenge the subtractive underpinnings that 

permeate schools serving bilingual communities (Cummins, 2000; García et al., 

2008; Khisty & Willey, 2008; Valencia, 2002). Laura’s experience, in which a 

teacher educator worked with her to incorporate discursively rich mathematics in-

struction in English and Spanish, is not the norm. Much more attention in teacher 

education and the mathematics education research community must be placed on 

the learning opportunities of bilingual students and bilingual teachers. Those who 

prepare bilingual teachers for the classroom must also model teaching practices that 

honor prospective teachers’ language and cultural knowledge because how they are 

taught paves the way for the possibilities they see to reimagine mathematics learn-

ing in their future classrooms.  

 
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Journal of Urban Mathematics Education Vol. 10, No. 2  138 

Appendix A 

 

Codes – Organized Mathematics Learning for Emergent Bilingual Students 

I. Language Support 
a. Teacher direct modeling (e.g., repeated use of word, emphasizing 

pronunciation, sophisticated use of language, simplifying the lan-

guage) 

b. Revoicing  
c. Student created products (e.g. word bank, definition, strategy) 
d. Gestures 

i. Student 
ii. Teacher  

e. Student analysis (e.g. compare words, strategies, word problem con-
text) 

f. Encouragement of L1 usage 
g. Visual aids (e.g., word banks, student strategies, concept maps) 
h. Games 
i. Use of real-life context 
j. Modify existing math textbook  
k. Focus on meaning making, not just students’ production of “correct” 

Spanish or English  

l. Ask students to analyze (e.g. compare words, concepts or meanings) 
II. Knowledge Authority 

a. Teacher presents solution method 
b. Teacher provides specific tool for problem solving 
c. Teacher has final word about correct answer/solution 
d. Multiple forms of student mathematical contributions are encour-

aged and valued  

i. Incomplete ideas 
ii. Incorrect answers 

iii. Multiple approaches to problem solving 
iv. Multiple representations  

e. Teacher encourages students to have final word about correct an-
swer/solution 

f. Teacher intentional selection of student contribution to minimize 
status among students (and specific subgroups) 

g. Position students to use one another as mathematical resource 
III. Language Practices 

a. Spoken words 
i. Spanish 

ii. English 



 

 

 

 

Yeh                                                                               Math Is More Than Numbers 

Journal of Urban Mathematics Education Vol. 10, No. 2  139 

iii. Hybridity  
b. Written communication (e.g. writing on white boards, student writ-

ten strategy, poster of student strategy) 

c. Symbolic/Equations 
d. Physical  

i. Math tools  
ii. Manipulatives 

iii. Realia 
iv. Objects 

e. Gestures  
i. Teacher 

ii. Student 
f. Contextual  

i. Teacher-generated 
ii. Student-generated