CREATING A POSSIBLE WORLD WITH ACTUAL MINDS ...

CREATING A POSSIBLE WORLD WITH ACTUAL MINDS: BILINGUAL STUDENTS AND EL MAGA, A TEACHER-LIKE CYBER FIGURE IN A MATHEMATICS AFTER-SCHOOL PROGRAM Higinio Domínguez University of Illinois at Chicago This study analyzes the message board of an after-school mathematics program theoretically rooted in Cole & Engeström’s (1997) cultural-historical activity theory, and Lave & Wenger’s (1991) communities of practice. Using the construct of third space and possible worlds, I traced the co-construction of an intimate and sustainable space, a possible world characterized by reciprocal heteroglossia (a sustained, responsive coexistence of different and differing voices) and epistemological intimacy ((the kind that destabilizes how participants come to know something). This paper raises an important question: If real students and a teacher-like figure were able to co-construct a space in which they reciprocally explore mathematics learning, can real teachers and real students co-construct and sustain a similar space in real mathematics classrooms? I. Introduction Research consistently reports that linguistic minority students are more likely to receive mathematics instruction of lesser quality than their linguistic majority peers (Faltis, 1996; Baker, 1988, 1993; Hornberger, 1991). This poor instructional quality is often associated to a like quality of teacher preparation programs. This paper considers the relationship between teacher and students as a possibility for improving the quality of mathematics teaching and learning. Using two theoretical constructs, third space and possible worlds, I analyze the message board of an after school program where students and El Maga, a teacher-like figure, exchanged messages for a year and a half. Analysis of these messages show that students and El Maga constructed a third space, but more importantly, sustained this space, transforming it into a possible world in which they experienced education not as a sure thing, but as an exercise of human speculation and negotiation (Bruner, 1986). I also contrast two concepts associated with learning: identity and identification, illustrating with examples why identification is a more appropriate concept in this case. This paper shows that changing teacher-student relationships can significantly change the quality of mathematics education of linguistic minority students. II. Conceptual Framework To make sense of the data for this study, I looked for theoretical constructs from various thinkers. Common across these ideas is the capacity of human beings to imagine possibilities and to inhabit these possibilities in ways that transform their world of experiences. A. Third Space I started with Gutierrez et al’s (1995) construct of third space and Bruner’s (1986) idea of possible worlds. While Gutierrez et al’s (1995) theorize about what it takes to produce a

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space for teachers and students to interact about learning interests without letting personal positions or agendas interfere with this genuine learning, they show how ephemeral this third space tends to be, as participants quickly recede back to more comfortable positions that they know well and therefore feel comfortable in them. Bruner (1986), on the other hand, writes about possible worlds that resemble the third space but in a more sustainable fashion. According to Bruner, “it is far more important, for appreciating the human condition, to understand the ways human beings construct their worlds (and their castles) than it is to establish the ontological status of the products of these processes” (p. 46). Bruner’s idea led me to look for more ideas related to making the third space a more durable space. In the next section I explain how these ideas contribute to sustain a third space. B. Theoretical Constructs for Thinking about a Sustainable Third Space Echoing Bruner’s (1986) possible worlds is Holland et al.’s (1998) figured worlds. “A figured world is peopled by figures, characters, and types who carry out its tasks and who also have styles of interacting within, distinguishable perspectives on, and orientations toward it” (p. 51). It is important to note that a world that has been “figured” is more than just a world that exists in people’s imagination. It actually means a world realized from the power of a collective imagination, an imagination of human beings who want live in worlds that transcend the simple and ordinary. In this sense, a figured world is a possible world, not a dreamed world but a place or space that allows people to make meaning and in turn gives meaning to people’s actions. Through this construct, Holland et al. (1998) remind us that our actual worlds and what we do in them, from worrying about how we look to writing academic books are in fact fueled by how we imagine ourselves and the importance we give our actions in these worlds. While Gutierrez et al. (1995) note that a third space tends to be short- lived, Holland et al’s (1998) explain that by virtue of continued participation and appropriation, “the world itself is also reproduced, forming and reforming in the practices of its participants” (p. 53). From a social-historical perspective, Vygotsky (1978) also theorizes about the power of imagined worlds by looking at how child’s play “seems to be invented at the point when the child begins to experience unrealizable tendencies” (p. 93). That is, children after going through that early period when they can have their needs satisfied immediately, they begin to develop new needs, such as the need to become someone or experiment with something new. This is when they, according to Vygotsky, invent play: when they realize that “Imagination is a new psychological process” (p. 93) that they can use to enter worlds otherwise not within their immediate reach. To enter these new worlds, it is not uncommon that the child uses a pivot, such as a stick or some object that represents say a horse. They also negotiate rules for these games, which makes their activity rule governed. Finally, Bakhtin’s (1981) theory of dialogic imagination also helps understand, from a perspective that focuses on the historical life of discourse (and power), how human beings experience the process he called “ideological becoming.” This process, according to Bakhtin, consists of selective appropriation of other people’s words or discourses. The process unites actuality with possibility in that “language … is never unitary” (p. 288)

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and “of all the words uttered in everyday life, no less than half belong to someone else” (p. 339), which encapsulates the factual part of this duality. The possibility aspect of this ideological becoming is associated to the internally persuasive discourse, which “is not finite, it is open; in each of the new contexts that dialogize it, this discourse is able to reveal ever newer ways to mean” (p. 346). Because the ideological becoming is based on the historical life of discourses, I prefer to address the issue of identity as identification, since a year and a half of data is a relatively short time compared to students’ longer histories of contact with various discourses. In the next section I explain what I mean by identification. C. Identification in and with Possible Worlds The concept of identity has been discussed in association with mathematics learning (Nasir, 2002; Nasir & Saxe, 2003). However, because identity evokes answers about who you are, it is rather too ambitious to claim that students in this after school program are as a result of a year and a half of participation in the program. Therefore, a more appropriate concept is identification, which evokes answers about who they want to be with or what kind of learning do they wish to be experimenting with. Here’s one example from a recent personal experience. In a recent visit to bookstore in Chicago that specializes in architecture and design, I first read about it on its website which described the place as: “serving those interested in architecture and design, whether students, professionals, or general enthusiasts.” In my case, the last phrase “general enthusiasts” validated not my identity as an architect or designer, which I am not, but my identification with these fields. Boaler & Greeno (2000) discuss the importance of developing in students, especially minorities, a sense of identification with the field of mathematics for the purpose of attracting more minorities to careers in mathematics. D. Relevance of Theoretical Concepts to Teacher-Student Relations The ideas previously discussed are relevant for thinking about teacher-student relationships, particularly in terms of envisioning a process of re-signification by which language minority students would experience education in a meaningful way. Thus, the question guiding this paper is: How can teachers and students not only produce a third space, as suggested by Gutierrez et al. (1995), but more importantly how can they sustain this space long enough to make it a possible world (Brunner, 1986), or a figured world (Holland, 1998)? A Model for Creating and Sustaining a Possible World

Teacher discourse

Student discourse

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In this model, the circles are initially separated, as teachers and students inhabit their own private spaces. The third space (Gutierrez et al., 1995) or the contact zone (Bakhtin, 1981) has not been created.

Teacher discourse

Student discourse

Third space

When the circles intersect, a third space is created, indicated by the intersection of the teacher discourse and student discourse. In Gutierrez et al’s (1995) analysis, this third space does not last long, as participants return to their private spaces. According to Gutierrez et al’s (1995) analysis of discreet episodes of teacher-students interactions, the creation of third space is a short-lived classroom phenomenon for which teacher and students are responsible as they opt to recede back to their own spaces after a quick moment of cohabiting this space. So what makes a third space become a durable space for dialog? It is the participants focus on the intersection. It takes imagining and believing that this space is a co-construction that provides a forum for ideas to emerge. It takes identification with the space as providing membership. It takes acting in this space in a continued manner. The space exists as long as the participants want it to exist. It takes, in more mundane words, giving “maintenance” to these intersections. In the following sections I explained how the creation and maintenance of a third space was possible in an after school program. III. Participants The study uses data from an after school program conducted in an urban Chicago public school with a large proportion of Latino bilingual students. Although the school’s official bilingual program is dual language education, many students express resistance to use Spanish in their mathematics learning interactions. The data consisted of one and a half year of message exchanges between the students and El Maga, a character borrowed from La Clase Mágica (Vasquez, 2003). In many ways, El Maga behaved and was perceived by students as a teacher-like figure. For example, El Maga (a) encouraged students to be curious about mathematics, (b) fostered respect for other people’s ideas, (c) answered students questions selectively, (d) asked students questions (both mathematical and nonmathematical), (e) students came to El Maga for one-to-one interactions, they trusted her, and (f) was a figure with authority. IV. Data Analysis A year and a half of message exchanges permitted a view of the third space as it is constructed and developed over time. But besides the longitudinal perspective on third space, not explored by Gutierrez et al (1995), my paper is interested in how to make this

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space sustainable. To study this dimension, I started a process of open coding (Miles & Huberman, 1994) by reading all the messages in the message board and creating purely descriptive codes, that is, descriptions of what participants communicated in their messages without any further interpretation. But once data were coded completely, the codes suggested the possible existence of several intersections. These intersections, I claim, are responsible for the sustainability of a third space, or rather the continued attention to them by participants is what makes a third space sustainable. The codes are organized in four major areas: (A) Codes About Establishing Rules for Writing, (B) Codes About Students’ Identification with Mathematics, (C) Codes About Students Identification with Language, and (D) Codes About Students Identification with a Possible World. VI. Findings In this section I present the codes and illustrate them with at least one example of how students’ discourse and El Maga’s discourse intersected, creating and maintaining a focus on this possible world. Original text is reproduced as students typed it, sometimes with misspellings. A. CODES ABOUT ESTABLISHING RULES FOR WRITING (a) Participants Co-construct and Adhere to Rules for Writing Messages The first rule that participants negotiated was about the use of writing for relevant experiences. They adhered to this rule as illustrated by this exchange:

El Maga writes: I was hoping to get news from you, but it didn't happen? How come? I hope you write back soon! EL MAGA

Student responds: NOTHING TO SAY

(b) Students Write About Personally Meaningful Experiences Students dedicated this space for writing about what mattered to them in their lives. They could do this because their messages were not judged or discouraged by El Maga. For example, one student wrote: Student writes: Hola el Maga El juego que jugue fue de dominoes y mis vacasiones fue chidas. Tu ves deportes? como futbol. Si o no? Y si lo ves, cuales jugadores te gustan de la liga de futbol mexicana. A mi me gusta chivas por que ganaron y son los campiones y si no me gustaran a mi entonces no serian campeones.

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Translation: Hi El Maga The game I played was dominoes and I had a great vacation. Do you watch sports? Like soccer? Yes or no? If you do, which player from the Mexican soccer league do you like? I like Chivas because they won and they are the champions, and id I didn’t like them they wouldn’t be champions.

B. CODES ABOUT STUDENTS’ IDENTIFICATION WITH MATHEMATICS (c) El Maga Asks Students Mathematical Questions These kinds of messages started after the students and El Maga had time to know each other more personally via the message board. For example, a student asked El Maga the question 81 divided by 9 and El Maga modeled a problem solving strategy (derived facts), asked the student for an alternative strategy, and sent a new problem to the student. El Maga writes: Yo amo las matematicas, pero hay veces me olvido como hacerlo, aver... 81 dividido por 9...9 por 4 son 36, 9 por 7 son 63, 9 por 10 son 90, aha! 90 menos 9 son 81, entonces 81 dividido por 9 son 9. Tu sabes una forma mas rapida para sacar la respuesta? Ten, yo te voy a dar unos problemas ahora:

Translation: I love mathematics but sometimes I forget how to do it, let’s see…81 divided by 9…9x4 is 36, 9x7 is 63, 9x10 is 90, a-ha! 90 minus 9 is 81, so 81 divided by 9 is 9. Do you know a faster way to get the answer? Here, now I am going to give you some problems: What’s 54 divided by 9?

que son 54 dividido por 9? (d) Students Ask El Maga Mathematical Questions Students reciprocated El Maga’s messages with more mathematical questions and demanded more mathematics problems from El Maga. The third space became populated with mathematical problems in a competitive yet friendly exchange. Student writes: Buenas tardes, Querido maga te voy hacer un problema eddie compro 23 manzanas. Le dio 12 a su hermana. cuantas manzanas que daron en total. Me mandas tambien un problema.

Translation: Good afternoon, dear Maga, I am going to give you a problem. Eddie bought 23 apples. He gave 12 to his sister. How many apples did he have left? You send me a problem too.

(e) Students and El Maga Negotiate Views on Mathematics Freely, students expressed their views about mathematics, influenced by their experiences in school with this subject. In this example, a student associates coming to school with doing boring mathematics. El Maga invites students to transform his views. Student writes: So when are you relli coming to are shool? When you come are you going to tell us boring math ?

El Maga responds: Hi Alfredo, I'm glad that you are with us now. I really enjoy seeing you working with your friends. It is difficult for me to come and visit you because I'm from the fifth dimension that's why I have the Under Graduates to work with you. I talk and send them messages just as I do with you. 6

I'm glad because they like Mathematics and working with boys and girls. And yes, you could have boring Math if you want, but the most interesting Math is that that makes you think and solve problems and situations that are interesting or real. I'm planning to give you a mission and start finding that interesting Mathematics around you. C. CODES ABOUT STUDENTS’ IDENTIFICATION WITH LANGUAGE

(f) Students code-switch Students communicated their experiences in the language they used in those experiences, as in this example. Student writes: Dear Maga:

Translation: Dear Maga:

I am so sorry that you do not celebrate Valentine's Day in the fifth dimension. And yes I did like the chocolates that you gave for Valentine's Day. At the mechanics we looked at the motor of a truck. It was a little bit messy because of all the grease from all the cars. Hoy yo estuve escribiendo preguntas para preguntarle cuando vallamos paratras. I wanted you to say hi to Christina for me, I really miss her. It is ok that you had not written back to me and the good things is that you are writing to me right now. I am not mad at you and I hope that you have a lovely day.

I am so sorry that you do not celebrate Valentine's Day in the fifth dimension. And yes I did like the chocolates that you gave for Valentine's Day. At the mechanics we looked at the motor of a truck. It was a little bit messy because of all the grease from all the cars. Today I was writing questions to ask (the mechanic) when we go back next time. I wanted you to say hi to Christina for me, I really miss her. It is ok that you had not written back to me and the good things is that you are writing to me right now. I am not mad at you and I hope that you have a lovely day.

(g) Creative Use of Language Students are becoming bilingual in English and Spanish. Here’s an example, quite common in their writing, of Spanish phonology used in English spelling. Note that El Maga responds to the meaning of the message, not to the student’s misspelling. Student writes: Hi Helomaga i am fane

El Maga responds: Hello G (student’s initial) I am happy to hear that you are doing fine.

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(h) Students Evaluate El Maga’s language In fact, El Maga’s gave students the opportunity to “try on” roles not commonly experienced in most classrooms, such as the opportunity to correct El Maga’s English and Spanish as in these examples.

Student writes: HELLO, I had chichen and pasta at my party. You messed up writing the letter when you wrote what did you did over the weekend. Anyway, I didn't do much on the weekend. We worked on a bubble gum project today and the mother of the twins would of spent four cents on the three colored gum ball machine. I gotta go to Tae Kwon Doe. See you later. Holla el maga tu escribiste pollitos mal se escribe p o l l i t o s no pollilos.Pero no inporta yo me la pase bien en las matematicas.El error de tu dinero fue que estabas contando por 4 hasi que tuviste un error en 24 y en vez de 24 tenia que ser 26.

Translation:

Hello Maga, you wrote chickens wrong. It is spelled “pollitos” not “pollilos”. But that’s alright, I had a good time in mathematics. The mistake with your money is that you were skip-counting by 4, so you made a mistake in 24, and instead of 24 it was 26.

(i) Students Use Language to Impart Imaginative Meanings to Experiences Students inhabited worlds that they constructed with their imagination. These worlds were mathematical. In this example, the student refers to the situation in a combinations problem as if it was the real event of packing clothes for a vacation. Student writes: Querido maga, perdoname pero si te escribi no entiendo por que no llego.Te escribi un bonito mensage y si esto bieny te deje unas caritas de sonrrisa al final de la carta. Entonces te quiero decir loque hice ayer. Estuvimos empacandoropa para ir de vacaciones conelmaga. Y solamente teniatres blusas ydos pantalones y descubri que hay 6 formas de vestirce.

Translation: Dear Maga, forgive me but I did write to you, I don’t understand why you didn’t get it. I wrote you a nice message and I am fine and I even put some smiling faces at the end of my letter. Anyways, I want to tell you what I did yesterday. We were packing clothes to go on a vacation with El Maga. And I only had 3 blouses and 2 pants and I found out that there are 6 ways to get dressed.

In this example, a student sees a geometric figure as a friend.

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Student writes: DEAR Maga I made a new imaginary friend from tooth picks and marshmallows.

In this example, a student imagines himself as being a chess player. Student writes: HELLO MAGA TODAY I PLAYED ALOT OF CHESS.WHEN I GROW UP IM THINKING OF BEING A CHESS PLAYER.I PLAYED WITH MARCOS TWO TIMES AND I BEAT HIM BOTH TIMES.I LIKE PLAYING CHESS.DO YOU LIKE CHESS.I LOVE CHESS.I LIKE MY HOMEWORK BECAUSE WE ARE SUPOSE TO DRAW A PICTURE FOR ONE THING WE WONDER.I SAID THAT I WONDER IF I COULD BE A GOOD CHESS PLAYER.DO YOU THINK THAT I COULD BE A GOOD CHESS PLAYER.RIGHT NOW MARCOS IS PLAYING THE COMPUTER. Finally, two students asked mathematical questions in the form of wonderments about the meaning of familiar mathematical concepts, such as multiplication and age, but in the future. Since El Maga lived in a 5th dimension, students associated El Maga with the dimensions existing in the future. Student asks: Holla maga cuantos anos tienes en el futuro. Holla maga Que senefica 43x23 en el futuro.

Translation: Hello Maga, how old are you in the future? Hello Maga, what does 43x23 mean in the future?

D. CODES ABOUT STUDENTS IDENTIFICATION WITH THIS POSSIBLE WORLD (j) Students express desire to stay connected On occasions that El Maga did not respond immediately to students’ messages, students demanded a prompt reply. Initially, the student in the following example had spent considerable time asking El Maga to reveal his/her gender. At this point, the student is more concerned about maintaining communication than the information she was trying to obtain. Student writes: DEAR MAGA porque no me escribiste ase mucho que no me as escribido desde

Translation: DEAR MAGA, why you haven’t written to me? It’s been a long time since you wrote,

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since last week. Please write back to me.

la semana pasada.porfavor escribeme para tras.

(k) Students Express Sense of Achievement/Challenge/Expertise Within the world that students and El Maga co-constructed and co-inhabited, students felt confident about their mathematical achievements.

Student writes: Hola Maga

Translation: Hi Maga

Te exttrane este verano.Leimos tu carta y de tu gatito. Me puedes decir algo de tu gatito como elcolor. Hoy trabajamos con fracciones y resolvimos las fracciones y decia 'Fracction Power.' Ahora me gustan mas las fracciones que antes. Saque el segundo lugar de mi mesa. Hasta pronto Maga

I missed you this summer. We read your letter and about your little kitty. Can you tell me something about your kitty, like the color. Today we worked with fractions and we solved them and it said ‘Fraction Power.’ I like fractions better now than before. I got the second place of my table. So long Maga

(l) Students Affiliate to Mathematics/Language/World Knowledge as Identity Markers Finally, in this possible world, students showed important affiliations to mathematics, language, and world knowledge as marking of what they identify with in this space. Students writes: Hello my name is A (student’s initial). I want to get too know you more often. Do you know spanish En mi fin de semana yo fui al parque. Fui al parque y juge futbal con mi papa y mi mama. Juegos en los monkey bars. Uso mi bike afuera de mi house. Yo se multiplicar y dividid y escribir. Hice como hacer una pizza con dos ingredientes. Yo me confudi con en vez de decir peperoni dije pepino. Ja ja.

Translation:

This weekend I went to the park. I went to the park and played football with my dad and my mom. We played at the monkey bars. I use my bike outside the house. I know how to multiply and divide and write. And I know how to make a pizza using two ingredients. I got confused and instead of saying pepperoni I said cucumber. Ha ha.

Discussion

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In this study, I used theoretical constructs such as the third space (Gutierrez, et al, 1995), possible worlds (Bruner, 1986), figured worlds (Holand et al., 1998) and the ideological becoming (Bakhtin, 1981) to make sense of the complex but communication between bilingual students and El Maga in an after school program. Although El Maga shares some characteristics with real classroom teachers, something that El Maga did that is not so common among teachers was subordinate teaching to learning. This aspect of El Maga’s interaction with students made him/her vulnerable. But only through this subordination could El Maga experience the learning of mathematics from the students’ perspectives, by being in the contact zone (Bakhtin, 1981) where another’s discourse is a double-voiced discourse. This aspect in El Maga-students interactions also engendered what I called reciprocal heteroglossia and epistemological intimacy. From the beginning of the program, students expressed curiosity about El Maga’s undefined gender. Although El Maga downplayed this inquiry, students’ curiosity did not disappear completely. This is a productive intersection in that curiosity, in this case expressed about gender, is characteristic of children. What teachers do with this desire to know more is crucial for the education of children, especially children who have been historically disadvantaged in education. Children soon came to understand that there were other concerns of more importance than asking about El Maga’s gender, as exemplified in the codes. Another productive intersection between El Maga’s and the students’ discourse was the use of imagination evidenced in the many of the mathematical questions asked. This illustrates Bakhtin’s (1981) dialogic imagination in that students infused events otherwise trivial such as age with imaginative dimensions (what is your age in the future?). They, in other words, voluntarily stepped off their own reality to enter El Maga’s reality and ask questions from El Maga’s perspective. How often do teachers do something similar to this act of courage these students demonstrated? Students also used references to language, mathematics and world knowledge as markers of identity for themselves as well as for others with whom they interact. These identity markers are key moments that teachers must focus on because they are like X-rays of this complex construct we refer to as identity. And just like doctors use X-rays to determine what’s wrong with one’s health, teachers can use this information to make a difference in how students are socially forming their identities or identifications as bilingual learners. Students and El Maga exchanged roles as they invented this imagined space. For example, students corrected El Maga’s errors in English and Spanish, a role more typical of teachers than of students. They were learning to talk to traditional authority figures such as a teacher-like figure, but they were also talking to learn important ideas about mathematics (Wells, 1986). Finally, this study illustrates how transforming education requires transforming the figured world that it is into the figured world that it could be. Such transformation implies significant changes in participants’ affinities and identifications. In this paper I have suggested that one way to produce such change is by examining teacher-student

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relationships, interactions, how teachers orient themselves to students, how students orient themselves to teachers, what they talk about to each other, what one does with what the other says or does, in short, how they inhabit a world that is transformable only by them. For language minority students, the concept of re-imagining a world, in this case the world of mathematics, is crucial as a way of helping them re-imagine themselves as capable mathematical thinkers. Author’s Note The preparation of this paper was supported in part by a grant from the National Science Foundation to the Center for the Mathematics Education of Latinos (No. ESI-0424983). The findings and opinions expressed here are those of the author and do not necessarily reflect the views of the funding agency. References Baker, C. (1988). Key issues in bilingualism and bilingual education. Clevendon: Multilingual Matters. Baker, C. (1993). Bilingual Education in Wales. In H. Baetens Beardsmore (Ed.), European Typologies of Bilingual Education. Clevendon: Multilingual Matters. Bakhtin, M. M. (1981). The dialogic imagination. Austin TX: The University of Texas Press. Boaler, J. & Greeno, J. G. (2000). Identity, agency, and knowing in mathematics worlds. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning, pp. 171-200. London: Ablex Publishing. Bruner, J. (1986). Actual minds, possible worlds. Cambridge, MA: Harvard University Press. Cole, M., Engeström, Y., & Vasquez, O. (1997). Mind, culture, and activity: Seminal papers from the laboratory of comparative human cognition. New York: Cambridge University Press. Faltis, C. (1996). Learning to teach content bilingually in a middle school bilingual classroom. Bilingual Research Journal, 20(1): 29-44. Gutierrez, K., Rymes, B., & Larson, J. (1995). Script, counterscript, and underlife in the classroom: James Brown versus Brown v. Board of Education. Harvard Educational Review (65)3: 445-471. Holland, D., Lachicotte Jr., W., Skinner, D., and Cain, C. (1998). Identity and agency in cultural worlds. Cambridge, MA: Harvard University Press. Hornberger, N. H. (1991). Extending enrichment bilingual education: Revisiting typologies and redirecting policy. In O. García (Ed.), Bilingual education:

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Focusschrift in Honor of Joshua A. Fishman (Vol. 1). Amsterdam/Philadelphia: John Benjamins. Lave, J. & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. New York: Cambridge University Press. Miles, M., & Huberman, M. (1994). Qualitative data analysis: An expanded sourcebook. Thousand Oaks, CA: Sage. Nasir, N. (2002). Identity, goals, and learning: Mathematics in cultural practice. Mathematical Thinking and Learning (4)2&3: 211-245. Nasir, N. S., & Saxe, G. B. (2003). Ethnic and academic identities: A cultural practice perspectve on emerging tensions and their management in the lives of minority students. Educational Researcher (32)5: 14-18. Vasquez, O. A. (2003). La clase mágica: Imagining optimal possibilities in a bilingual community of learners. Mahwah, NJ: Lawrence Erlbaum Associates. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. M. Cole, et al. (Eds.) Cambridge, MA: Harvard University Press. Wells, G. (1986). The meaning makers: Children learning language and using language to learn. Portsmouth, NH: Heinemann.

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