“ Explícame tu Respuesta ”: Supporting the Development of Mathematical Discourse in Emergent Bilingual Kindergarten Students

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“Explícame tu Respuesta”: Supportingthe Development of MathematicalDiscourse in Emergent BilingualKindergarten StudentsSylvia Celedón-Pattichis a & Erin E. Turner ba University of New Mexicob University of ArizonaPublished online: 13 Aug 2012.

To cite this article: Sylvia Celedón-Pattichis & Erin E. Turner (2012) “Explícame tu Respuesta”:Supporting the Development of Mathematical Discourse in Emergent Bilingual Kindergarten Students,Bilingual Research Journal: The Journal of the National Association for Bilingual Education, 35:2,197-216, DOI: 10.1080/15235882.2012.703635

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Bilingual Research Journal, 35: 197–216, 2012Copyright © the National Association for Bilingual EducationISSN: 1523-5882 print / 1523-5890 onlineDOI: 10.1080/15235882.2012.703635

“Explícame tu Respuesta”: Supporting the Developmentof Mathematical Discourse in Emergent Bilingual

Kindergarten Students

Sylvia Celedón-Pattichis

University of New Mexico

Erin E. TurnerUniversity of Arizona

This study investigated Spanish-speaking kindergarten students’ participation in mathematicaldiscourse as they solved and discussed a range of word problems. Specifically, we draw uponsociocultural perspectives on mathematics learning to frame mathematical discourse and to exam-ine specific teacher and student actions that seemed to support the development of mathematicaldiscourse over the course of the kindergarten year. Data sources included pre- and post-task-basedclinical interview assessments and weekly (videotaped) observations of problem-solving lessons.Findings demonstrated ways that teachers supported and students appropriated discursive habits suchas using more precise mathematical language, explaining solutions in ways that referenced actionson quantities in the problem, and using multiple visual representations to mediate communication.In addition, the findings point to the critical role the teacher plays in supporting the development ofstudents’ mathematical discourse.

INTRODUCTION

Much prior research on young children’s problem solving has been from a cognitive tra-dition, focusing on students’ mathematical thinking, their use of specific strategies, andtheir understanding of particular problem structures (Carpenter, Ansell, Franke, Fennema, &

Both authors contributed equally to this manuscript.Sylvia Celedón-Pattichis is Associate Professor in the Department of Language, Literacy, and Sociocultural Studies at

the University of New Mexico and was a Co-PI of the Center for the Mathematics Education of Latinos/as (CEMELA).Her research interests include linguistic and cultural influences on the teaching and learning of mathematics with Latina/ostudents, especially with second language learners.

Erin E. Turner is Assistant Professor of Mathematics Education in the Department of Teaching, Learning andSociocultural Studies at the University of Arizona. Her research interests include issues of equity and social justicein teaching and learning mathematics, with a particular focus on teaching mathematics in culturally and linguisticallydiverse contexts.

Address correspondence to Sylvia Celedón-Pattichis, College of Education, Department of Language, Literacy, andSociocultural Studies, MSC05 3040, 1 University of New Mexico, Albuquerque, NM 87131-0001. E-mail: [email protected]

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198 CELEDÓN-PATTICHIS AND TURNER

Weisbeck, 1993; Carpenter, Fennema, Franke, Levi, & Empson, 1999; Carpenter, Hiebert &Moser, 1981). While this research has led to significant advances in our understanding of youngchildren’s mathematical thinking, it has tended to foreground particular aspects of children’sproblem solving, such as the sophistication of different strategies (Carpenter et al., 1999), whilebackgrounding others, such as the complex ways that children leverage a variety of cultural,social, linguistic, or material resources to solve problems and communicate their reasoning.This traditional lens is particularly problematic for students from nondominant groups, such asLatina/o students, English learners, and emergent bilingual students.1 For instance, a cognitivelens typically does not emphasize how children draw upon cultural and linguistic knowledgeand experiences to support their participation in school mathematics (e.g., González, Andrade,Civil, & Moll, 2001; Moschkovich, 2002), which may decrease opportunities to highlight theresourcefulness of students’ contributions, and in doing so, inadvertently contribute to deficitframings of students (Moschkovich, 2010).

We advocate a sociocultural perspective on mathematics learning. This perspective viewslearning as an inherently social and cultural activity, where “participants bring multiple views to asituation; representations have multiple meanings for participants; and that these multiple mean-ings for representations and inscriptions are negotiated through conversations” (Moschkovich,2002, p. 197). A sociocultural lens also frames learning mathematics as akin to learning toparticipate in mathematical discourse practices such as justifying thinking, articulating solutionstrategies, and posing questions to others (Moschkovich, 2002, 2007a; O’Connor & Michaels,1996; Sfard, 2001). Framing mathematics learning in this way is consequential for students fromdiverse cultural and linguistic backgrounds, because it draws our attention to the multiple under-standings and resources that students leverage to communicate their ideas and how teachers mightdraw on these resources in instruction (Moschkovich, 2002).

In this study, we examine young children’s problem-solving activity through a socioculturallens, with a particular focus on how children learn to participate in mathematical discourse.Specifically, we investigate the following research questions: How does a teacher support thedevelopment of emergent bilingual Latina/o kindergarten students’ mathematical discourse asthey solve and discuss word problems across the kindergarten year? What resources do studentsdraw upon as they begin to take up particular discursive habits?

CONCEPTUAL FRAMEWORK

Mathematical Discourse and Mathematical Discourse Practices

For at least two decades, there has been growing consensus that understanding how and whatstudents learn in school mathematics requires attending to students’ opportunities to communi-cate mathematical ideas and participate in mathematical discourse (Moschkovich, 2002, 2007a;NCTM, 2000; O’Connor & Michaels, 1996; Yackel & Cobb, 1996). For example, Sfard (2001)argued that, “we can define learning [mathematics] as the process of changing one’s discursiveways in a certain well-defined manner” (p. 25). Researchers have conceptualized mathematical

1We draw from the work of García and Kleifgen (2010) to capitalize on students’ linguistic and cultural backgroundsas resources and view students as capable of acquiring English and maintaining their native language as they engage inchallenging mathematical tasks.

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SUPPORTING MATHEMATICAL DISCOURSE IN EMERGENT BILINGUALS 199

discourse in different ways. Some have focused on precise mathematical terminology (Spanos,Rhodes, Dale, & Crandall, 1988), while others have attended more broadly to the complex waysthat students use formal and everyday language, home language, and code-switching betweenlanguages to communicate mathematical reasoning (e.g., Setati, Adler, Reed, & Bapoo, 2002).

Still other research has sought to distinguish between language and discourse, arguing thatmathematical discourse is more than the language that students use to voice their ideas. Forexample, Moschkovich (2002, 2007b), drawing on Gee’s (1996) work on Discourses as waysof speaking, being, thinking, and interacting, explained that mathematical discourse includesoral language, as well as symbolic representation, tools, gestures, and ways of knowing, com-municating, and behaving. Our study draws on this latter understanding of discourse where“language is a tool, whereas discourse is a broader activity in which the tool is used” (Gutiérrez,Sengupta-Irving, & Dieckmann, 2010, pp. 45–46).

Furthermore, we understand mathematical discourse to be socially constructed and situated inthe practices of particular communities (Moschkovich, 2010; Willey, 2010). Discourse commu-nities such as elementary mathematics classrooms are often characterized by particular “ways ofbeing, thinking and speaking that are unique to the mathematics environment” (Willey, 2010,p. 531), such as specific ways of explaining ideas, negotiating definitions, and symbolizingsolutions (McClain & Cobb, 2001). Various studies, which we discuss in greater detail, havedescribed how “experts (teachers) gradually induct novices (students) into the[se] desired formsof thinking, reasoning, and valuing” (O’Connor, 1988, p. 24).

Mathematical Discourse Practices and Young Bilingual Students

Mathematical Discourse in Kindergarten Classrooms

While we know that young children can and do participate in mathematical discourse(McClain & Cobb, 2001), research with children as young as kindergarten age is limited.Moreover, these studies have tended to focus on the teacher’s conceptions about the importanceof mathematical talk (Jung & Reifel, 2011) and to a lesser extent, general practices that teach-ers use to support mathematical discourse, such as encouraging opportunities to communicatemathematical ideas (Cooke & Buchholz, 2005; Jung & Reifel, 2011). Less frequently examinedhas been how specific teacher moves support developments in kindergarten children’s discourseover time.

Mathematical Discourse in Bilingual Classrooms

Even less explored is how young emergent bilingual students develop mathematical discoursepractices. One reason for the limited research is that early studies of bilinguals’ mathematicalactivity focused on more limited views of learning, such as acquiring vocabulary and com-prehending word problems (Mestre, 1988; Spanos et al., 1988) or understanding the multiplemeanings of words across mathematical and everyday contexts (Moschkovich, 2002). Anotherreason is that emergent bilingual students are often excluded from linguistically demandingactivities such as explaining and justifying ideas and instead participate in more procedu-ral tasks (Iddings, 2005; Planas & Gorgorío, 2004). This exclusion reflects an erroneousassumption that when a student is learning the language of instruction, s/he is unprepared to

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participate in mathematical conversations. More recent research has countered this assump-tion by documenting how bilingual students can and do participate in mathematical discourse,sometimes using informal, everyday language, and drawing upon metaphors, objects, gestures,and code-switching as resources to communicate ideas (Domínguez, 2005; Moschkovich, 1999,2002; Setati, 2005).

Supporting the Development of Mathematical Discourse

Research has also attended to the teacher’s role in supporting the development of mathematicaldiscourse. For example, Sfard (2000, 2001) uses the metaphor of a pump to describe how teach-ers support the expansion of students’ mathematical discourse, by intermittently introducing newsymbols, names, representations, and discursive habits that students can then take up to commu-nicate understanding and generate new ideas. Specifically, Sfard draws attention to how teachersfacilitate shifts in students’ use of (a) mathematical vocabulary, (b) visual mediators to communi-cate ideas (i.e., symbols, representations), as well as (c) changes in the meta-discursive rules thatguide students’ mathematical conversations (i.e., norms for mathematical justifications; see alsoMcClain & Cobb, 2001). Related to Sfard’s notion of meta-discursive rules, Yackel and Cobb(1996) described how primary grade teachers and students established norms for what counts asacceptable mathematical explanations and justifications, as well as criteria for mathematicallydifferent solution methods. Consistent with these norms, over the course of the year, childrenincreasingly explained their strategies by describing specific actions on quantities in a problemrather than by retelling a set of procedures disconnected from the problem context (see alsoMcClain & Cobb, 2001).

More specific to linguistically diverse learning environments, Hufferd-Ackles, Fuson, andSherin (2004) studied the development of mathematical discourse in a third grade classroomwith significant numbers of Latina/o and English language learners. They found that studentexplanations progressed from one-word statements of answers to short phrases and later morecomplete and confident descriptions of reasoning as the teacher began to probe students’ thinkingmore deeply, thereby reinforcing meta-discursive rules that require explanations. Other researchin linguistically diverse settings has found that teachers can support mathematical discourse byencouraging students to amplify their communication with multiple resources, including visualmediators such as symbols and drawings (Turner, Domínguez, Maldonado, & Empson, 2010),by modeling rich mathematical talk, including mathematical vocabulary (Khisty & Chval, 2002),and by focusing on the mathematical content of students’ contributions, rather than their accuratepronunciation or grammar (Moschkovich, 1999).

Lacking in prior research is a focus on the development of mathematical discourse inclassrooms with young, emergent bilinguals. In our own research we have documented howa kindergarten dual-language teacher (Spanish/English) helped Latina/o, Spanish-dominantstudents make sense of a range of basic word problems, using strategies such as posingproblems in an informal storytelling manner and grounding tasks in familiar, relevant con-texts (Turner & Celedón-Pattichis, 2011; Turner, Celedón-Pattichis, & Marshall, 2008; Turner,Celedón-Pattichis, Marshall, & Tennison, 2009). The present study takes a more socioculturallens, focusing explicitly on teacher and student moves that seemed to support the developmentof young emergent bilinguals’ discourse over the course of the kindergarten year. Specifically, we

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attend to development of three dimensions of mathematical discourse highlighted in our frame-work above: students’ use of mathematical vocabulary; students’ use of visual representations,including symbols and tools, to mediate their communication; and meta-discursive norms aboutappropriate forms of participation.

METHODOLOGY

Setting and Participants

Participants for this study included the teacher (Ms. Arenas) and students from one kindergartenclassroom in a school with predominantly Latina/o student populations (87%), where almostall students (over 90%) qualified for free or reduced lunch. Ms. Arenas, also a native Spanishspeaker, followed a 90-10 bilingual model of instruction (Lindholm-Leary, 2001); almost allmathematics lessons were in Spanish. We selected this classroom because Ms. Arenas hadparticipated in professional development focused on young children’s mathematical thinking(Cognitively Guided Instruction [CGI]: see Carpenter et al., 1999), and she was interested inconducting problem-solving lessons with her students and encouraging mathematical discourse.Although Ms. Arenas had extensive experience teaching mathematics to young children, this washer first year using a CGI framework to understand children’s thinking.

All of Ms. Arenas’s students were dominant Spanish speakers (as determined by the Pre-LAS2

Assessment Scale: De Ávila & Duncan, 1998) with varying degrees of English-languageproficiency. Students’ performance on kindergarten readiness assessments (KindergartenDevelopmental Progress Record [KDPR]: Albuquerque Public Schools, 2005) reflected a rangeof understandings typical of entering kindergartners. Most students could rote count to 10, andslightly more than half could recognize at least some numerals between 1 and 10. Other studentsbegan the year with less-developed number skills (see Turner & Celedón-Pattichis, 2011).

Problem-Solving Lessons in Ms. Arenas’s Classroom

Problem-solving lessons formed a core component of Ms. Arenas’s mathematics instruction.She began presenting problem-solving tasks at least twice a week in mid-October and continuedthrough the remainder of the year. In a typical lesson, Ms. Arenas presented a word problemorally, often framed as a story about herself or other members of the classroom community. Shethen encouraged students to solve the problem in ways that made sense to them, offering a varietyof tools such as counters and cubes, number lines, and whiteboards to support their reasoning.After students solved the problem, Ms. Arenas facilitated a group discussion of students’ strate-gies. A typical lesson lasted 20 minutes and involved solving and discussing a range of problems(e.g., addition, multiplication, compare problems).

2The Pre-LAS is a measure of students’ proficiency in Spanish and English as well as preliteracy skills from 5 to7 years of age.

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Data Collection

Classroom Observations

Beginning in mid-October, we visited Ms. Arenas’s classroom on a weekly basis to observeand videotape problem-solving mathematics lessons. Over the course of the year, we videotapeda total of 25 lessons, all of which Ms. Arenas taught in Spanish to reflect the 90-10 model usedin the school. Some lessons involved small groups (i.e., Ms. Arenas sitting at a table with fiveor six students, posing and discussing several problems), and others were whole-class lessons.For this study, we selected 10 lessons for closer analysis, using the following three criteria.First, given our interest in the development of mathematical discourse over time, we selectedlessons from different parts of the school year (three from November/December, three fromJanuary/February, and four from March/April). Second, we chose lessons to reflect the dis-tribution of whole- and small-group lessons across the year (i.e., Ms. Arenas worked almostexclusively with small groups during the fall semester, and thus lessons selected from this timeperiod involved small groups. Beginning in January, Ms. Arenas used whole-group instructionmore frequently, and thus four of the seven lessons from the spring semester were whole group).Third, given our focus on discourse, we selected lessons where the audio and video data were ofhigh enough quality to produce clear and complete transcripts.

Pre- and Post-assessments

To complement classroom observation data, and to compare students’ mathematical dis-course from the beginning to the end of the year, we conducted pre- and post-clinical interviewassessments (Ginsburg, Kossan, Schwartz, & Swanson, 1983) with eight of the 22 students inMs. Arenas’s classroom. This sample included four girls and four boys and was selected to rep-resent a range of achievement levels on a beginning-of-the-year kindergarten readiness measure(KDPR). We administered the preassessment in early October, before Ms. Arenas began regularproblem-solving lessons, and the postassessment in May.

Each assessment included problem-solving tasks modeled after the problems used in theCarpenter et al. (1993) study (see the appendix). All problems were presented orally in students’dominant language (Spanish), and students were told that interviewers could read problems ineither language (Spanish/English), as many times as needed. All but one of the eight studentschose to maintain Spanish as the language of the interview. At the beginning of each interview,researchers set out a range of tools (counters, cubes, paper and pencil) and informed studentsthat they could use these tools to support their thinking. Consistent with our focus on mathemat-ical discourse, after each problem children were invited to explain their strategy and to clarifyand justify different parts of their solution. Interviewers asked questions such as, “How did youfigure out the problem? What did you do first?,” among others. Assessments were videotapedand transcribed for analysis. Transcripts attended to students’ oral language as well as how theyleveraged tools and representations (e.g., pictures, gestures) to communicate their ideas.

Data Analysis

To investigate the development of students’ mathematical discourse, we analyzed transcriptsfrom pre/postassessments and problem-solving lessons with attention to three dimensions of

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mathematical discourse discussed in our conceptual framework (McClain & Cobb, 2001; Sfard,2001; Yackel & Cobb, 1996): (a) the mathematical language used by the child and/or teacher;(b) the visual means by which students communicated their thinking, as well as ways thatteachers invited students to mediate communication with a range of visual representations (e.g.,invitations to share thinking using tools, modeling of explanations that involve gestures); and(c) how students’ contributions and teacher’s responses (i.e., follow-up probes, revoicing withrefined language) reflected meta-discursive rules about explaining and justifying one’s reasoning.

For each dimension, we developed a set of codes that reflected key aspects of mathe-matical discourse. For example, for the mathematical language dimension, codes includedeveryday language, mathematical terminology, and teacher modeling of mathematical language,among others (Khisty & Chval, 2002; Moschkovich, 2007b). Codes for the second dimensionattended to visual representations students (or teachers) used to communicate ideas such asgestures, symbols, drawings, charts, concrete objects (tools), counting on a number line, etc.(Domínguez, 2005; Moschkovich, 1999; Sfard, 2001). Finally, codes for the third dimensionwere used to mark instances where teachers initiated or students evidenced particular discur-sive habits, such as expectations to justify answers mathematically, to explain strategies in waysthat referred to the problem context, and to use mathematical terminology (Yackel & Cobb,1996).

All transcripts were coded by two members of the research team, and differences were dis-cussed until agreement was reached. Consistent with our goal of understanding how teacher andstudent moves may have supported the development of students’ mathematical discourse, anal-ysis focused on relationships between teacher actions and student contributions. The followingsections outline the findings and implications of our research.

FINDINGS

In the sections that follow, we describe features of emergent bilingual students’ mathemati-cal discourse early in the kindergarten year, drawing on preassessment data. Next, we exploredevelopment in students’ discourse across the year, noting teacher and student moves thatseemed to support these shifts. We end with a portrait of students’ mathematical discourseat the end of the year. To illustrate these findings, we use interview and lesson episodesinvolving four students (two girls—Dalia and Araceli, and two boys—Gerardo and Emilio)whose mathematical discourse reflected the range and development noted in our broader set ofdata.

Mathematical Discourse Early in the Kindergarten Year

At the beginning of the year, students’ explanations often consisted of short phrases that restatedanswers or procedures (i.e., “It’s 3” or “I counted”)3 without further justification. While teachers(and interviewers) intended to elicit explanations grounded in the quantities and mathematicalrelationships in the problem, students’ responses typically did not evidence this discursive habit.Instead, they seemed to abide by a different set of discursive norms, ones where “seeing the

3All student responses were in Spanish. Any examples provided in English are translations.

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answer” and “just knowing it” were suitable justifications.4 For example, after using cubes tofigure out that if he had two pockets with two coins in each pocket, he would have four coinsaltogether, Gerardo was asked to explain his solution. He stated that he knew four was the answerbecause he “put four” (line 3) and “because it was four” (line 5).

1. G: Cuatro. (“Four.”)2. Int5: Cuatro. ¿Cómo supiste que cuatro? (“Four. How did you know it was four?”)3. G: Porque . . . porque poní cuatro y . . . (“Because . . . because I put four and . . . ”)4. Int: Está bien. A ver. Dime cómo supiste que cuatro. (“It’s fine. Let’s see. Tell me how

you knew it was four.”)5. G: Porque si era como cuatro y estaba cuatro, pos . . . pos cuando lo quieres poner en

otro número, va a ser otro número. (“Because if it was like four and there was four,well . . . well when you want to put it [the answer] on another number, it’s going to beanother number.”)6

We should note that Gerardo’s explanation was mediated by his use of physical tools (i.e.,cubes) to model the problem. That said, other than saying “I put four [cubes]” he did not referto the tools in his explanation, either orally or through gestures (i.e., pointing to the sets ofcubes). For example, he did not use the cubes to describe quantities in the problem (i.e., twocubes represent two coins in each pocket), nor how he operated on those quantities to find theanswer (i.e., setting out two cubes for one pocket, two cubes for the other pocket, and countingall cubes to find the total). This suggests that while visual representations were part of Gerardo’smathematical discourse, he did not leverage those tools to describe his solution strategy.

In other instances, students’ explanations evidenced attempts to describe actions taken to solvea problem. For example, after repeated probes from interviewers to describe their strategies, somestudents seemed to realize that “It just is [the answer]” was not a sufficient justification. Students’next move was often to describe an action taken to solve the problem, typically “I counted.”While in most cases “I counted” was an accurate description of part of their strategy, studentsalso used “I counted” to justify solutions not based on counts (e.g., a guess).

For example, Dalia listened to a basic addition problem (Jennifer has six candies and her sistergives her three more, how many does she have now?) and quickly announced that “eight” wasthe answer. While it was unclear how Dalia arrived at “eight” (she did not visibly count or modelthe problem), based on the fact that she struggled to solve most of the preinterview tasks and thatwhen asked to restate this problem she simply said that Jennifer had six candies and then eight,we suspect that “eight” was a guess. When probed to explain her solution (line 2), Dalia said sheknew it was eight “porque conté” (line 3), and when asked to show how she counted, she countedout eight cubes without reference to the quantities in the problem (line 5).

1. D: Ocho. (“Eight.”)2. Int: ¿Cómo supiste? (“How did you know?”)

4Although students’ explanations were limited at the beginning of the year, they demonstrated an impressive capacityto reason about and solve word problems, which we describe in greater detail in other reports (see Turner et al., 2008,2009; Turner & Celedón-Pattichis, 2011).

5“Int” refers to the interviewer. “G” refers to Gerardo.6It is difficult to translate informal Spanish to informal English without violating certain cultural emphases and

meanings.

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3. D: Porque conté. (“Because I counted.”)4. Int: A ver, ¿me puedes enseñar cómo contaste? (“Let’s see, can you show me how you

counted?”)5. D: [She gets the cubes and counts them up to eight—Uno, dos, tres . . . ocho.]6. Int: Ah, ¿Pero cómo supiste que tenía ocho? (“Ah, but how did you know she had eight?”)7. D: Pensé en mi mente. (“I thought in my mind.”)

One interpretation of why Dalia claimed that she counted (when she likely had guessed) isthat she recognized “I counted” as a viable response to a probe to explain one’s thinking. Daliamay have heard other students explain their solutions using this phrasing, and so her responsemay reflect an emerging awareness of meta-discursive rules in her classroom. Similarly, wesuspect that Dalia responded to the interviewer’s request to show how she counted by draw-ing on a common practice in her kindergarten classroom, counting out a set of objects. Sheseemed to use this practice as a template for her response, similar to Sfard’s (2001) descriptionof how students use previously developed discursive habits as templates to reason about newideas. In other words, for Dalia “show me how you counted” did not mean to explain how oneused counting to act upon the quantities in the problem; it simply meant to count. In line 7,when Dalia realized that the interviewer wanted more than “I counted,” she shifted to anotherphrase typical of students’ early in the year discourse, Pensé en mi mente (“I thought in mymind.”).

While students’ discourse was often characterized by restatements of answers, and shortphrases describing that they “just knew,” a few students offered explanations that did referencethe quantities and actions in the problem (McClain & Cobb, 2001) and that included visual meansof communicating mathematical ideas (Sfard, 2001). For example, to solve a basic addition prob-lem (“I had four candles and then got three more; now how many do I have?”), Araceli raisedfour fingers and then counted up three more to get seven. She explained that she knew there wereseven because she counted on her fingers (line 2), and in response to a further probe (line 3),demonstrated her counting on strategy (line 4).

1. Int: ¡Qué bien! ¿Cómo supiste? (“Great job! How did you know?”)2. A: Los conté con mis dedos. (“I counted them with my fingers.”)3. Int: ¿Los contaste? ¿Me puedes enseñar cómo los contaste? (“You counted them? Can

you show me how you counted them?”)4. A: Puse cuatro [showing four fingers], y luego conté uno, dos, tres [puts up one finger at

a time]. (“I put four [showing four fingers], and then I counted one, two, three [puts upone finger at a time].”)

Unlike Dalia, Araceli used “counting” to demonstrate how she started with one quantity inthe problem and then added on the second quantity (line 4). Additionally, she used a visualrepresentation (her fingers) to mediate her explanation.

To summarize, early in the kindergarten year, students’ mathematical discourse was char-acterized by restatements of answers and short statements that typically did not reference thequantities or mathematical relationships in the problem. However, there was also evidence thatstudents were using visual representations (i.e., cubes, fingers) to mediate their discourse, even ifthey typically did not leverage those representations to support or clarify their explanations. In thenext section we use three extended episodes to illustrate developments in students’ mathematical

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discourse across the kindergarten year. In each episode, we highlight how both Ms. Arenas andher students contributed to this development.

Students’ Participation in Mathematical Discourse Across the Kindergarten Year

Episode 1: Emilio Explains His Strategy for a Subtraction Problem

During a lesson in early November, Ms. Arenas posed (in Spanish) a subtraction problembased on a recent class art activity (“We made six paintings, three got wet, how many are stilldry?”). Emilio solved the problem by counting out six cubes, separating three, and counting thosethat remained. As he described his solution (line 2), he said “he thought” there were six, typical ofphrasing students used on preassessments. Ms. Arenas then suggested a different discursive habit,encouraging Emilio to leverage his visual representation (the cubes) to support his explanation(line 3).

1. Ms. A: ¿Cómo lo hizo? (“How did you do it?”)2. E: Pensaba, yo pensaba que era seis. [He has six cubes on the table, starts to pick up

three]. (“I thought, I thought it was six.”)3. Ms. A: Uh-huh, seis, enséñame los seis. (“Uh-huh, six, show me the six.”)4. E: Y se mojó [picks up three cubes] tres. (“And they got wet [picks up three cubes] three.”)5. Ms. A: Muy bien, tú pensaste que eran seis, y luego tres se mojaron, entonces los quitaste,

¿no? (“Very good, you thought there were six, and then three got wet, then you took thoseaway, no?”)

6. E: [nods]7. Ms. A: Muy bien. Y ¿te quedaron cuántos, mi amor? (“Very good. And, how many were

you left with, my love7?”)8. E: Tres. (Three.)

This episode demonstrates how, at Ms. A’s encouragement, Emilio was beginning to usevisual representations (cubes) and reference to quantities and actions in the problem to frame hisexplanations (“And then they got wet, three” line 4). In this way, Ms. A’s probes and restatementscoupled with Emilio’s responses served to establish (and reinforce) meta-discursive rules aboutexplaining one’s mathematical thinking (i.e., explanations needed to reference quantities in theproblem and to leverage visual representations). Additionally, we see how Ms. Arenas modeledmathematical language that Emilio could use to describe his actions on the quantities ( . . . “thenyou took those away, no?” line 5, “And, how many were you left with?” line 7). Ms. Arenasused similar moves—modeling mathematical language and inviting students to use visual repre-sentations to scaffold their explanations—throughout the year, and we suspect that such movescontributed to the development of students’ mathematical discourse.

7Ms. Arenas often referred to students as mis amorcitos (“my little loved ones”).

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Episode 2: Araceli and Gerardo Describe Solutions for a Multiplication Problem

In a whole-group lesson from February, Ms. Arenas posed a multiplication problem basedon the upcoming Valentine’s Day holiday: Ms. Arenas bought three bags of chocolate hearts,and each bag had three hearts. “How many hearts do I have in all the bags?” Araceli solvedthe problem by drawing three bags and three circles in each bag, and then counting the circlesto find the total. When asked to share her solution with the class, Araceli began by retellingthe story (lines 2 and 4), supporting the norm that explanations should reference the problemcontext.

1. Ms. A: Vamos a pensar. A ver Araceli. (“Let’s think. Let’s see Araceli.”)2. A: Compraste tres bolsitas. (“You bought three little bags.”)3. Ms. A: Escuchen a Araceli. (“Listen to Araceli.”)4. A: Y luego tenía tres corazones . . . (“Then you had three hearts . . . ”)5. Ms. A: Tres corazones en cada bolsita. (“Three hearts in each bag.”)6. A: Y son nueve. (“And there are nine.”)

Ms. Arenas responded by providing a more precise way of describing the number of heartsper bag (tres corazones en cada bolsita, line 5). In doing so, she modeled for Araceli thatparticipating in mathematical discourse involved precise mathematical descriptions.

As the interaction continued, Ms. Arenas pushed Araceli to further explain her answer(line 7). Ms. Arenas offered another discursive habit—describing the actions one takes to reacha solution—as a potential template for Araceli’s explanation (lines 7 and 11). Araceli seemed totake up this suggestion, and began to use her drawing, and her actions on that drawing (lines 8and 10), as tools to communicate mathematically.

7. Ms. A: Okay. ¿Cómo? Muy bien. ¿Qué hiciste para saber? (“Okay. How? Very good.What did you do to know?”)

8. A: Puse los corazones. (“I put the hearts.”)9. Ms. A: Muy bien. (“Very good.”)

10. A: Y los conté. (“And I counted them.”)11. Ms. A: Muy bien. Cuente para nosotros. (“Very good. Count for us.”)12. A: Uno, dos, tres, cuatro, cinco, seis, siete, ocho, nueve (“One, two, three, four, five, six,

seven, eight, nine) [Pointing to the hearts as she counts].”

Notable here is the contrast between Araceli’s use of counting (line 12) and how Dalia countedin a prior example. Whereas Dalia counted out a set of objects to “justify” a guess, Aracelicounted the nine circles (hearts) in her drawing to demonstrate how she found the solution (i.e.,Araceli’s counting was connected to actions on quantities in the problem).

This excerpt highlights Ms. Arenas’s efforts to support the development of Araceli’s math-ematical discourse. By modeling refined versions of Araceli’s statements (line 9), and probingAraceli to explain her solution in ways that referenced visual representations and actions onquantities in the problem (line 7, line 11), Ms. Arenas encouraged shifts away from old dis-course habits (i.e., state the answer) toward new discourse habits (Sfard, 2001). In this way,Ms. Arenas’s moves, coupled with Araceli’s responses (i.e., describing and modeling actions),functioned to support the development of mathematical discourse.

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Next, Ms. Arenas invited Gerardo to share. Like Araceli, Gerardo solved the problem bydrawing three bags and placing three tallies in each bag, and counting all the tallies to find thetotal. He then wrote an (incorrect) equation to represent his solution: “3 + 3 = 9.” Gerardostarted by stating, “I was thinking” (phrasing reminiscent of the preassessment). But then hequickly shifted to describe how he modeled the bags of hearts on his whiteboard (lines 1 and 3),reinforcing the discursive norm that explanations should reference actions on quantities in theproblem. When Ms. Arenas pushed him to justify the answer (line 4), he referred to his equa-tion, which served as a visual mediator of his discourse (line 5). Ms. Arenas then focusedGerardo’s attention on how his equation connected to the quantities in the problem (lines 8and 12).

1. G: Como yo estaba pensando . . . so tení . . . por estaba poniendo tres en tres en cadabolsita. (“Like I was thinking . . . so I have . . . I was putting three by three in each littlebag.”)

2. Ms. A: Muy bien. Y sacó . . . (“Very good. And you got . . . ”)3. G: Son nueve. (“There are nine [pointing to the 9 in his white board].”)4. Ms. A: Muy bien. ¿Cómo supiste que son nueve? (“Very good. How did you know there

are nine?”)5. G: Porque yo pu- . . . como yo poní el . . . yo poní el tres más tres son nueve. (“Because

I placed . . . like I placed the . . . I put three plus three is nine [He traces the numbers andthe symbol as he says the equation].”)

6. Ms. A: Pero tres más tres . . . son ¿cuántos? (“But three plus three . . . is how many?”)7. Another S: Seis. (“Six.”)8. Ms. A: Seis. ¿Qué le faltó ahí, Gerardo? Porque lo dibujó muy bonito. ¿Cuánto tiene cada

bolsita? ¿Cuánto tiene cada bolsita? (“What are you missing there, Gerardo? Becauseyou drew very nicely. How many does each little bag have? How many does each littlebag have?”)

9. G: Tres. (“Three”)10. Ms. A: Muy bien. Y ¿cuántas bolsitas son? (“Very good. And how many little bags are

there?”)11. G: Tres. (Three.) [He is counting the 3s in the equation but still says tres.]12. Ms. A: Pero ahí pusistes abajo en los números sólo pusiste dos bolsitas . . . mira . . . de

tres. (But you put there below in the numbers you just put two little bags . . . look . . . ofthree.) [referring to 3 + 3] [Gerardo looks at the equation.]

13. Ms. A: ¿Cómo puedes agregar otra bolsita de tres? (“How can you add another little bagof three?”)

14. G: [Writes another + 3 to get 3 + 3 + 3 = 9.]

In this excerpt, Ms. Arenas used various moves to support the development of Gerardo’smathematical discourse. First, by probing Gerardo to reconsider whether three plus three is actu-ally nine (line 6), and whether his equation included all quantities in the problem (“What areyou missing there?” line 8), she communicated a discursive norm that symbolic representationsneed to be accurate and to reflect one’s solution strategy. Next, she drew Gerardo’s attention tothe number of bags in the problem (lines 10–11) and noted that he needed to represent each ofthese bags in his equation (“But there below in the numbers you just put two little bags . . .

of three,” line 12). By pointing to his equation (3 + 3 = 9) as she referenced the problem

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context, we suspect that she helped Gerardo connect the symbols to the objects in the prob-lem. Gerardo seemed to make sense of this idea, adjusting his equation (adding an additional“+ 3”) to generate an accurate representation (line 14). In this way, Gerardo’s actions rein-forced the discursive norm that visual representations should accurately reflect one’s solutionstrategy.

Episode 3: Dalia Describes Her Solution to a Multistep Problem

This final excerpt is from a whole-group lesson in April. During this lesson, Ms. Arenasposed several problems related to the upcoming Easter holiday, including the following multistepproblem: “Ms. Arenas had two boxes of chocolate eggs, and each box had four eggs. Then I atetwo chocolate eggs. How many do I have left?” Dalia, who solved the problem by drawingtwo boxes with four small circles in each box, and then crossing out two circles and countingthose that remained, shared her solution first (line 1). As she began to retell the story (line 2),Ms. Arenas prompted her to focus on the quantities in the problem (line 3) and then modeled howshe might use her visual representation to describe her actions on those quantities (line 5 and 7).

1. Ms. A: A ver, Dalia, tú no has pasado. Ven a explicar. (“Let’s see, Dalia, you havenot come up. Come explain.”) [Dalia goes up to the front of the room and takes herwhiteboard.]

2. D: Primero Ms. Arenas se comió . . . (“First, Ms. Arenas ate . . . ”)3. Ms. A: ¿Cuántos había en cada cajita? (“How many were there in each box?”)4. D: Cuatro. (“Four.”)5. Ms. A: Okay, muy bien, y pusiste cuatro en cada cajita. (“Okay, very good, and you put

four in each box.”) [She has drawn two small squares, with four circles in each square.]6. D: (nods)7. Ms. A: Y luego me comí dos. (“And then I ate two.”)8. D: y le quedaron seis. (“And you were left with six.”)9. Ms. A: ¿Y cómo supiste? (“And how did you know?”)

10. D: Porque los conté, cuántas bolitas quedaron. (“Because I counted them, how manylittle balls were left.”)

11. Ms. A: ¿Y las otras? ¿Cómo sabes cuáles son los que comí? (“And the others? How doyou know which ones are the ones I ate?”)

12. D: Estas. (“These.”) [points to two circles that are crossed out]

In this episode, Ms. Arenas used a series of probes (lines 3, 9, and 11) to elicit details ofDalia’s strategy and to communicate the discursive norm that one needs to explain one’s thinking.Additionally, she provided a frame for Dalia’s explanation that was based on describing theactions in the story (Y luego me comí dos, line 7) and explaining how she represented thoseactions in her drawing (¿Cómo sabes cuáles son los que comí?, line 11). We also see how Daliaresponded in ways that reinforced important discursive norms—by using more mathematicallanguage (i.e., stating that six were left) and referring to visual representations (i.e., indicatingthat she crossed out circles in her drawing) to communicate her reasoning.

In summary, Ms. Arenas used a variety of strategies, throughout the kindergarten year, to sup-port the development of emergent bilingual students’ mathematical discourse. Most importantly,

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she continued to communicate the meta-discursive rule that when she invited students to explaintheir solution, they could not respond by stating, “I just knew.” Comments such as the followingstatement from a lesson in February were typical of her conversations.

1. Ms. A: No, pero no me digan, [yo] pensé. Quiero saber cómo lo hizo. . . . Miren aquí pues. Porquelo hicieron bien . . . . Pero me tienen que explicar cómo porque ustedes nada más me están diciendo‘pensé,’ y no me están explicando cómo. (“No, but don’t tell me ‘I thought.’ I want to know how youdid it . . . . Look here then. Because you did it right . . . . But you need to explain to me how becauseyou are only telling me ‘I thought,’ and you are not explaining how.”)

We suspect that Ms. Arenas repeatedly revisited this discursive rule because she recognizedthat taking on new discursive habits is challenging for students, a point we return to in the discus-sion. In the final section of our findings, we draw on postassessment data to summarize patternsin students’ mathematical discourse at the end of the year.

Mathematical Discourse at the End of the Kindergarten Year

Some aspects of students’ discourse at the end of the year mirrored the discourse students usedon the preassessment. For instance, students continued to use visual representations (i.e., cubes,drawings, etc.) to mediate their explanations, but they more often referenced these representationsas they described their solution strategies, as the following examples demonstrate. Additionally,in a few instances students continued to use phrasing like “I just knew,” but at the end of the yearthese statements often indicated that the student knew the addition or subtraction fact associatedwith the problem or that they had mentally counted to find the answer. For example, when askedto explain his solution to “You have six candies and then get six more. How many do you havenow?,” Gerardo responded:

G: Estaba pensando en mi mente, mi mente estaba pensando que era doce. Y yo también, y luego loconté . . . con mi voz de adentro. (“I was thinking in my mind, my mind was thinking that it was 12.And then I also, I also counted . . . in my voice inside [my mind].”)

Instances when students used “I was thinking in my mind” or “I just counted” to justifyguesses were infrequent and typically associated with more challenging problem structuresthat students struggled to solve. This suggests that students’ enhanced capacity to solve abroader range of word problems and to use more sophisticated problem-solving strategies alsocontributed to developments in students’ discourse (Turner & Celedón-Pattichis, 2011).

In general, students’ mathematical discourse at the end of the year was characterized by math-ematical descriptions of their thinking. In particular, students seemed to have appropriated themeta-discursive rule that solution strategies need to be explained in ways that reference the quan-tities and actions in the problems. For example, Araceli solved a division problem (“You have10 cookies and some bags. You want to put two cookies in each bag. How many bags do youneed?”) by drawing 10 tally marks and circling two tallies at a time to get the answer of five.

1. A: [She draws 10 tally marks and circles two at a time.] Cinco. (“Five.”)2. Int: ¿Cómo sacaste cinco? (“How did you get five?”)

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3. A: Puse diez galletas, luego puse bolsitas nomás con dos, y luego conté las bolsitas, yluego sabía que eran cinco. (“I put 10 cookies, then I put boxes only with two, and thenI counted the boxes and then I knew it was five.”)

In her explanation (line 3) Araceli connected her visual representation (circles around tallymarks) to the mathematical action of dividing a set of objects into groups of equal size. Her useof a visual representation both to support her solution strategy and to mediate her explanationmarked an important development in mathematical discourse.

In addition, at the end of the year students’ mathematical discourse was often characterizedby more precise mathematical language. For example, Dalia solved a Join Change Unknownproblem (“Ana has seven dollars. How many more does she need to buy a toy plane thatcosts 11 dollars?”) by drawing 11 tally marks, crossing out seven, and then counting those thatremained to get four. She then explained her thinking:

1. D: Primero Ana quiere comprar un avión que cuesta 11 dólares. Y le faltan . . . [countingthe tally marks she did not cross out] 1, 2, 3, 4. (“First Ana wants to buy a plane that costs11 dollars. And she needs [counting to the tally marks she did not cross out] 1, 2, 3, 4.”)

2. Int: ¿Cómo supiste que le faltan cuatro [dólares]? (“How did you know that she needsfour more [dollars]?”)

3. D: Puse 11, y quité los siete (“I put 11, and then I took away seven”) [points to the seventally marks that she crossed out], y luego le faltan cuatro (“and then she needs four”)[points to the four remaining tally marks].

Dalia’s explanation is typical of the kind of explanations that students provided on thepostassessment in that she referenced objects and quantities in the story (e.g., Ana quiere comprarun avión que cuesta 11 dólares), described how she operated mathematically on those quantities(e.g., Puse 11, y quité los siete), and used mathematical language that described relationshipsbetween quantities (e.g., Le faltan cuatro).

A final feature of students’ mathematical discourse at the end of the kindergarten year was thatsome students began to use symbolic notation to model their solution strategies. For example,Gerardo solved a subtraction problem (“Your friend David had 13 cookies. Then he ate five.How many cookies does David have left?”) by drawing 13 circles, grouping five of them, andthen counting those that remained to get eight. Like many students, he then justified his solutionby describing how he operated on the quantities in the story (line 1).

1. G: Cuando me dijiste que eran trece estaba contando así [shows 10 fingers on the desk,but indicates he was hiding them under the desk and was counting] . . . uno, dos, tres,cuatro, cinco, seis, siete, ocho, nueve, diez [then starts counting again]—once, doce, trece.Y luego cuando . . . cuando comió cinco, hice uno, dos, tres, cuatro, cinco [touching threefingers, then two on the other hand], y me salió ocho. (“When you told me there were 13,I was counting like this [shows 10 fingers on the desk, but indicates he was hiding themunder the desk and was counting] . . . one, two, three, four, five, six, seven, eight, nine,ten [then starts counting again]—11, 12, 13. And then when . . . when he ate five, I didone, two, three, four, five [touching three fingers, then two on the other hand], and I goteight.”)

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Next, the interviewer asked if he could use numbers to represent the solution, and he wrotethe number sentence, “13 – 5 = 8.” Gerardo then justified that this number sentence was accuratebecause in the problem, “they took away.” We see Gerardo’s use of symbolic notation to mediatehis explanation as an important marker of development of mathematical discourse.

In summary, these examples demonstrate how young emergent bilingual students’ math-ematical discourse developed across the kindergarten year. Specifically, students began toappropriate mathematics vocabulary, as well as visual and symbolic representations to communi-cate their reasoning (Sfard, 2001). Students also came to understand discursive rules that guidedproblem-solving discussions in Ms. Arenas’s classroom (Yackel & Cobb, 1996).

DISCUSSION

An important contribution of this study is illustrating how emergent bilingual Latina/o kinder-garten students leveraged a variety of resources—including tools, visual representations, actionson quantities, gestures, and symbols—to mediate and support the development of mathematicaldiscourse. We found that even at the beginning of the kindergarten year, students’ explanationsincluded some key elements of mathematical communication. For example, on the preassess-ments students leveraged physical tools to mediate their explanations, and in a few cases, theirexplanations referenced actions on quantities in the problem. In other instances, students’ dis-course suggested an emerging awareness of meta-discursive rules, as students attempted toappropriate discourse templates (e.g., “I counted”) they may have heard from peers. We seethe way that these young emergent bilinguals leveraged resources to support their mathemati-cal discourse, even at the beginning of the kindergarten year, as a response to Moschkovich’s(2010) call to provide examples of students from nondominant communities who successfullyengage in cognitively and linguistically demanding tasks. In addition, these findings challengeprior studies that have suggested that young children from various nondominant backgrounds(e.g., low socioeconomic status, African American and Latina/o students, bilingual students)enter school lacking in skills and experiences needed to engage in mathematical problem solvingand discussion (Clements & Sarama, 2007).

Another important contribution of this study is that it highlights specific teacher moves such asinviting students to use story contexts, tools, and visual representations to scaffold their explana-tions that supported developments in young emergent bilinguals’ mathematical discourse. Otherkey moves included modeling mathematical language, probing students for details about theirstrategies, and explicitly reminding students that statements such as “I just knew” were notacceptable explanations. Through these moves, Ms. Arenas introduced and reinforced impor-tant discursive habits such as explaining solutions in ways that referenced action on quantities inthe problem (McClain & Cobb, 2001). That said, our findings also suggest that developing newdiscursive habits is challenging (Sfard, 2001). Although Ms. Arenas modeled precise mathemati-cal language throughout the year, we found that some students persisted with vague explanations,even as other aspects of their mathematical discourse evidenced development. What seemed crit-ical in helping students to take up new discursive habits were persistent probes and modelingfrom teachers coupled with frequent opportunities for students to communicate their reasoningand to engage the reasoning of others. As Sfard (2001) argued, “whoever wishes to becomefully fluent in mathematical communication has to persist in practicing mathematical discourse”(p. 13). While many of the discursive strategies used by Ms. Arenas have been described in prior

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research, this study illustrates the specific ways that such moves were enacted with young emer-gent bilinguals. In this way, our findings extend the field’s understanding, in much needed ways,of the teacher’s role in promoting mathematical discourse.

We should also note that several other factors may have supported the development of mathe-matical discourse in Ms. Arenas’s classroom. Ms. Arenas had some understanding of CognitivelyGuided Instruction, and her knowledge of possible solution strategies in particular may havesupported her ability to probe students’ thinking and elicit their explanations. Another importantfactor in the success of these kindergarten students may have been that they learned mathematicsin their native language, which arguably enhanced their capacity to make sense of problems andcommunicate their reasoning (Domínguez, 2005; Moschkovich, 1999).

A final contribution of this study is its focus on how teachers and students worked togetherto support the development of mathematical discourse. While previous research has highlightedteacher moves, our study traces both the teacher’s actions (i.e., probes, gestures, models, restate-ments, etc.) and students’ contributions (i.e., use of tools, reference to story context or visualrepresentation). This lens of attending to both teacher and student actions not only allows us totrack developments in students’ discourse over time but also to highlight specific moves that mayhave contributed to this development.

CONCLUSIONS

This study documents how emergent bilingual Latina/o kindergarten students and their teacherinteracted to shift mathematical discourse as students solved and discussed word problemsover the course of the kindergarten year. Additionally, this study advances our understandingof how young emergent bilinguals actively participate in mathematics classrooms not only byreading and listening, but also by discussing, explaining, symbolizing, representing, justifying,and connecting mathematical ideas, all of which are critical to the success of these students(Application of Common Core State Standards for English Language Learners, 2010). We con-tend that students’ participation in these mathematical discourse practices was supported by theuse of meaningful and relevant problem-solving tasks and by teacher moves that invited studentsto take up new discursive habits. We argue that it is particularly important to support discoursepractices such as posing and challenging claims and questioning the reasoning of others withstudents from nondominant communities where the practice of questioning or critiquing others’work may not be a norm (Valdés, 1996).

We conclude with suggestions for future research related to young Latina/o students’ mathe-matics learning. Future studies might investigate the following questions, among others: (a) Howdoes emergent bilinguals’ mathematical discourse continue to develop as they enter upper gradelevels?, and (b) How do teachers support students’ mathematical discourse in multilingualsettings, and/or in settings where native language instruction is not feasible?

ACKNOWLEDGMENTS

This research was supported by the National Science Foundation, under grant ESI-0424983,awarded to CEMELA (The Center for the Mathematics Education of Latino/as). The viewsexpressed here are those of the authors and do not necessarily reflect the views of the fundingagency.

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We would like to thank other members of our research team, in particular Richard Kitchen,Edgar Romero, and Havens Levitt for their help in conducting pre- and post-assessment inter-views and classroom observations. We are especially grateful to Mary Marshall and AlanTennison who provided professional development and helped with data collection. Without theirhelp this Kindergarten Study would not have been possible. We also want to thank Mrs. Arenasand her students for welcoming us into their classroom and sharing their knowledge with us.

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APPENDIX

Selected Postassessment Items8

Problem Type Problem

Join Result Unknown a. Julio tenía seis galletas, y luego su hermana le dio seis galletas más. ¿Cuántas galletastiene Julio ahora? (“Julio had six cookies, and then his sister gave him six morecookies. How many cookies does Julio have now?”)

Separate Result Unknown b. Paola tenía 13 dulces y luego se comió cinco. ¿Cuántos dulces le quedan? (“Paola had13 candies and then she ate five of them. How many candies does she have left?”)

Multistep c. Javier tenía dos bolsas de canicas. Había cuatro canicas en cada bolsa. Luego, él diotres de sus canicas. ¿Cuántas canicas le quedan? (“Javier had two bags of marbles.There were four marbles in each bag. Then, he gave away three of his marbles. Howmany marbles does he have left?”)

Division w/ Remainder d. Quince niños quieren pintar. Ellos van a sentarse en las mesas, pero solamente cabencuatro niños en cada mesa. ¿Cuántas mesas van a necesitar para que todos los 15 niñospuedan pintar? (“Fifteen children want to paint. They are going to sit at tables, but onlyfour children can fit at each table. How many tables are they going to need so that all ofthe 15 children can paint?”)

8English and Spanish versions of the pre-/post-assessment items were created by Spanish speakers of the researchteam. We started off with readily available items from CGI, then multiple members from the research team modifieditems to keep consistency with the language used and the type of problem.

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Documents

“ Explícame tu Respuesta ”: Supporting the Development of Mathematical Discourse in Emergent Bilingual Kindergarten Students