Over the last thirty years, there has been growing interest in how children learn in

Transcription

1 Multimodal Problem Solving and Probability: The Counters Game Research Problem & Significance Over the last thirty years, there has been growing interest in how children learn in informal and out of school contexts primarily through the prism of discourse analysis of interaction. One of the primary rationales for this development is the rigidity of formal, classroom discourse structures. While there has been growing recognition within the mathematics education community for the need to consider discourse in the learning of mathematics and mathematization (Rogers, 2005); there has been a dearth of research on how children engage in mathematical discourses in informal and out of school spaces. Given the well documented achievement gap between Latinas/os and their mainstream counterparts, documenting learning in informal contexts such as an after school club, provides us with a unique opportunity to enhance our understanding of how children mathematize using multimodal mediational tools. Following the primarily literacy driven work of Gutierrez (1999), Vasquez (2003), and Cole (2006) and the underlying theoretical principles of CHAT that guide it; we set out to create a similar activity system, whereby bilingual, Latina/o children more fully draw on their toolkit to develop mathematical thinking and discourse practices through participating in play (see Khisty, 2001, 2004). We define mathematical thinking and mathematical discourse as those practices that are primarily used to discuss and understand quantities, shapes, spatial relations, and deductive/inductive reasoning. Similar to other Fifth Dimension projects (i.e. Las Redes, La Clase Magica), play activities are central to learning and development where they serve to create hybrid spaces for learning through the use of multiple modalities, languages, and other mediational tools (both material and

2 ideational). These informal learning contexts demand an examination through multiple perspectives not contained in a single strand of discourse research (Nasir and Saxe, 2003). By selecting the Counters game from a myriad of games played in the Los Rayos de CEMELA (the name of our after school club situated within an urban, Midwestern, Latino/a, immigrant setting) our investigation is guided by the following questions: 1) How do bilingual children draw on their repertoire of linguistic tools to negotiate meaning within the Counters game? 2) How do language ideologies mediate discourse practices and participant roles within the observed interactional frames? Setting and Ethnographic Context Fourteen to twenty third- and fourth- grade students have been voluntarily attending Los Rayos de CEMELA, the after-school club, for nearly two years. They meet twice a week for one and a half hour sessions. Within the sessions, students are allowed many freedoms to choose their activities and to dictate the course a project will take. This design is deliberately chosen by researchers to create a drastically different environment than what is typically found in the traditional classroom. The activities are designed to foster high-order thinking and reasoning skills. The Counters Game The math activity in this study focuses on probability concepts as represented in the Counters Game, one of several planned activities in the after-school project materials. The Counters Game requires that players create their own game board, consisting of a rectangular strip of colored construction paper with hand drawn boxes spread from left to right, with one of each of the numbers from 2 through 12 written in each space. Each player receives 12 "counters" (small plastic cubes) and places them on their own game

3 board, arranging them according to their own preference. For example, players may put several counters on the numbers 5, 6, and 7 while leaving other numbered spaces unoccupied. All other arrangements of the counters are allowed as long as all are placed on a numbered space on the game board. A pair of dice is passed between each player who then rolls them to get a number between 2 and 12. Each player then checks their game board to see if any of their counters are on that number. If so, one piece is removed and discarded. Then, another player takes a turn rolling the dice. Players take turns rolling the dice until one person has removed all of his or her counters to win the game. Next, players begin another round and once again arrange their counters. In this activity, we are interested in observing how each student arranges his or her counters. Specifically, one might ask, "Do students place their counters using strategies formulated by observing the frequency of particular numbers appearing during random throws of dice?" For example, a student may notice that the numbers 6 and 7 appear more frequently than other numbers such as 2 and 12. As a result, the student's rearrangement of game board counters may indicate strategies that the student uses to figure out how to maximize chances of winning the game. Conceptual Frameworks In this section we broadly outline four components that guide, inform, and frame the analysis 1) Cultural Historical Activity Theory 2) Discourse 3) Language Ideologies 4) Hybridity. Cultural Historical Activity Theory The significance and role of play in human learning and development and in particular how children engage with more capable peers to proceed through the zone of

4 proximal development is one of the central premises of the CEMELA activity system (Vygotsky, 1978; Nicolopoulou, 1993; Engstrom, 1999). Of particular importance is how the nature of game activity is rule governed, goal directed and participants through the process of mediation engage in joint-activity to accomplish the goals of an activity, namely to problem solve and make meaning (Wertsch, 1985; 1991). Another significant feature of the CHAT framework is the centrality of history (both synchronic and diachronic; Cole, 1996) toward understanding how participants shift and appropriate the discursive tools to participate in culturally appropriate ways (Rogoff, 1991) and in this case as framed on multiple activity levels (the Counters Game, CEMELA, the school, etc.). The remaining components of our framework (discourse, language ideologies, and hybridity) could be viewed as being situated within the larger CHAT framework as mediational tools through which participants transform themselves, the game, and the cultural artifacts at their disposal. Understanding and reflecting on the process of coconstructing meaning involves an analysis of the social process of learning and assumes that meaning does not reside solely in the mind of the learner. Learning is viewed as a situated activity, thus, the process of moving from legitimate peripheral participation toward full participation is a key construct that helps us look for role shifts, especially in terms of epistemic stance (expert-novice) (Lave and Wenger, 1991; Ochs, 1996). The notion of stance is useful for interpreting how participants position themselves toward the truth of her utterances (epistemic stance) and their emotional intensity (affective stance (Ochs, 1996). The transformation of these stances through participation in the Counters Game is of particular interest to our study.

5 We draw on language ideologies work (see Razfar, 2005) in order to understand the function and purposes of language use (i.e. code-switching) in relation to the values and beliefs of the participants as well as the larger sociopolitical context in terms of uses of linguistic and cultural mediational means that are non-standard and have a history of being subordinated in public spaces. CEMELA as an activity system and the Counters Game are intentionally and explicitly designed on the basis of language ideologies that encourage the use of all available linguistic and cultural tools to the participants which in this case includes Spanish, English, and hybrid language practices. When students are bilingual or English Language Learners (ELLs), another dimension to their discourse requires further consideration. In the process of negotiating meaning, ELLs often depend on their native language when participating in the discourse of a learning community. Their speech can contain a mix of both languages as they struggle to communicate their thoughts. This phenomenon has been referred to as hybridity (Gutierrez, Baquedano-Lopez, and Tejeda, 1999) and takes into consideration that multiple, diverse, and even conflicting mediational tools can create a productive zone of development for learning that integrates multiple perspectives. Further research suggests that it is necessary to consider how oral language functions in learning environments from a linguistic background in order to understand how to take advantage of opportunities to create scaffolds for ELLs in discursive practice (Barwell, 2005; Fillmore, 2000). One type of scaffold involves contextualized learning, where students participate in mathematical activities that are not characterized by traditional IRE structures. The Counters Game in the CEMELA after-school project represents one

6 example of contextualized learning that ideally wouldn t restrict participant language use in order to pursue the goals of the acitivity. Discourse The difference between formal and informal mathematics discourse is the move away from traditional procedures where the teacher and students appear to engage in mathematical activities that constitute following procedural instructions in order to reach an algorithmic recipe for solving abstract mathematics problems (Cobb, Wood, Yackel, and McNeal, 1992). Such procedural interactions typically involve students' verbalizations that are restricted to answering questions that implicitly encourage them to follow a pattern of discourse that involves teacher elicitation, student response, and teacher evaluation (IRE). Such a structure does not encourage students to negotiate mathematical meanings and practices. In contrast, situations that allow students and teachers to explore mathematics through co-construction of mathematical realities with manipulative materials as well as abstract mathematical objects yield discourse patterns that represent students' moment by moment negotiation of learning and understanding concepts. These interactions allow more space and time for students to expose their thinking and experiment with concepts. In the process, more experienced participants can use this extra information and discursive space not found in typical IRE interactions in order to introduce scaffolds unique to the learner's needs (Cazden, 2001; Mercer, 1995). Methods of Analysis and Findings Data Corpus

7 For the purpose of this analysis we analyzed approximately 220 minutes of videorecorded Counters game activity over a three week period. We chose to focus on the Counters Game because of its participant structure and because a successful strategy for winning is based on an understanding of higher-level math concepts surrounding probability. The game structure requires every member to pay attention to each turn of the game with each player receiving control of the dice. In terms of procedure, each player has equal status, an identical role, and identical operations to perform in order for the game to proceed. It is assumed that this activity may provide a context where participants are likely to display interactions that typically are either obscured or limited by traditional classroom formats. In searching for an activity to use as a unit of analysis, we first examined field notes and video logs for activities that engaged multiple levels of participants (parents, students, fellows, undergraduates UGs ) in collective activity focused on discovery of math concepts not typically found in elementary school arithmetic. There were three such activities centered on probability concepts but we have excluded two of them because, unlike the Counters Game, their participant structures required separation of tasks and roles. Also, the two excluded activities occurred before the Counters Game and we hoped to find references to this history within the selected videos. We isolated approximately 177 minutes of video containing combinations of all levels of participants (intergenerational, expertise) engaged in playing the Counters Game. Included are two minute segments showing whole-group instruction before participants break into smaller groups.

8 Coding Scheme Given our research focus and questions we developed codes that could be used to develop our understanding of mathematical problem solving as mediated by multiple modalities within the Counters Game. Our codes include 1) Language Choice 2) Codeswitching 3) Epistemic Stance 4) Assistance Strategies (with special emphasis on the use of questions, types of questions, and direction of questions) 5)Repair 6)Instances of disagreement between participants. To insure a higher degree of validity and inter-rater reliability, each of these codes was developed through discussions with three members of the research team and applied independently by members of the research team. Random instances were then selected and discussed to enhance the reliability of the codes. The entire data corpus was coded and recoded three times to achieve greater reliability and validity. Findings Through our analysis of the participation of the Counters Game we have found a complex, multilayered, and multimodal discourse practices mediating the participation of children playing the Counters Game. In this proposal we would like to highlight some of the global findings. With respect to language choice, we found participants switching between English and Spanish regularly and in purposeful ways (Over 500 instances of Spanish use, and 270 instances of English use, and nearly 100 instances of codeswitching or hybrid practices). Given the dynamic participation structure code-switching occurred systematically for the purposes of assistance, making tasks more comprehensible, asking questions, joking, inclusion, and sometimes exclusion of central and peripheral participants. Children often indexed there awareness of speakers and non-

9 speakers of a particular code by switching to accommodate understanding. The amount of Spanish use was substantially more than English (approximately 2:1) and this ratio increased over time as participants become more comfortable with each other and with the game. We also examined numerous instances of cognitive and discursive tension (nearly 250). Sometimes the tension related to fairness suggesting self-regulation of fairness within the game play and other times it coincided with participant attempts to repair or correct one another. The ensuing disagreements about the nature of the correction suggested how meaning making is never absolute and contested terrain. In the presentation, we plan on illustrating these themes with vignettes from our data. Barwell, R. (2005). Integrating language and content: issues from the mathematics classroom. Linguistics and Education, 16, Cazden, C. (2001). Classroom discourse: The language of teaching and learning (2nd ed.). Portsmouth, NH: Heinemann. Cobb, P., Terry; Yackel, E.; and McNeal, B. (1992). Characteristics of classroom mathematics traditions: An interactional analysis. American Educational Research Journal, 29(3), Cole, M. (1996). Cultural psychology: A once and future discipline. Cambridge, MA: Harvard University Press, Cole, M. and The Distributed Literacy Consortium, (2006). The fifth dimension: An afterschool program built on diversity. New York: Russell Sage Foundation. Engstrom et al (eds) (1999) Perspectives on Activity Theory, Cambridge: Cambridge University Press. Gutierrez, K., Baquedano-Lopez, P., and Tejeda, C. (1999). Rethinking diversity: Hybridity and hybrid language practices in the third space. Mind, Culture, and Activity, 6(4), Gutiérrez, K. & Rogoff, B. (2003). Cultural ways of learning: Individual traits or repertoires of practice. Educational Researcher, 32 (5),

10 Lave, J. and Wenger, E. (1991). Situated Learning. Cambridge, MA: Cambridege University Press. Khisty, L.L. (2004, July). Language diversity and language practice: Why should mathematics educators care? Paper presented at the Social and Political Thematic Afternoon Session of the 10 th International Congress for Mathematics Education, Copenhagen, DK. Khisty, L.L. (2001). Effective teachers of second language learners in mathematics. In M. van den Heuvel-Panhuizen, (Ed.). Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 3, Utrecht, The Netherlands: The Freudenthal Institute, Utrecht University. Martin, L., Towers, J., and Pirie, S. (2006). Collective mathematical understanding as improvisation. Mathematical Thinking and Learning, 8(2), Mercer, N. (1995). The guided construction of knowledge: Talk amongst teachers and learners. Tonawanda, NY: Multilingual Matters. Nasir, N. and Saxe, B. (2003). Ethnic and academic identities: A cultural practice perspective on emerging tensions and their management in the lives of minority students. Educational Researcher, 32(5), Nicolopoulou, A. (1993). Play, cognitive development, and the social world: Piaget, Vygotsky, and beyond. Human Development, 36(1), Ochs, E. (1996). Linguistic resources for socializing humanity. In J. J. Gumperz & S. C. Levinson (Eds.), Rethinking linguistic relativity (pp ). Cambridge: Cambridge University Press. Razfar, A. (2005). Language ideologies in practice: Repair and classroom discourse. Linguistics and Education, 16, Rogers, R., Mosley, M., Hui, D. and O'Garro-Joseph, G. (2005). Critical discourse analysis in education: A review of the literature. Review of Educational Research, 75(3), Rogoff, B. (1991). Apprenticeship in thinking: Cognitive development in social context. New York: Oxford University Press Sawyer, R. (2004). Creative teaching: Collaborative discussion as disciplined improvisation. Educational Researcher, 33(2),

11 Vasquez, Olga A. (2003). La clase magica: Imagining optimal possibilities in a bilingual community of learners. Mahwah, New Jersey: Lawrence Erlbaum Associates. Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes (M. Cole, V. John-Steiner, S. Scribner, & E. Souberman, Eds.). Cambridge, MA: Harvard University Press. Wertsch, J. (1985) ed. Culture, communication, and cognition. Cambridge, Eng: Cambridge University Press. Wertsch, J. (1991). Voices of the mind. Cambridge, MA: Harvard University Press, Yackel, E., Cobb, P., and Wood, T. (1991). Small-group interactions as a source of learning opportunities in a second grade mathematics classroom. Journal for Research in Mathematics Education, 22(5),

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