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Kimberly Cardimona, PhD Bloomsburg University Pennsylvania State University

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Question What type of instructional strategies can we use to promote…… Active engagement? Independent problem solving? Academic vocabulary?

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A Mathematical Problem Common misconception Polled teachers, administrators, students: Believe math is universal Requires minimal reading/literacy skills Can be accomplished with minimal English language proficiency skills

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REALITY Mathematics requires reading and literacy skills: directions, word problems, operations Mathematics requires knowledge of arithmetic vocabulary Mathematics is culturally loaded/contain many cultural references (measurement, weight, time) Problem solving processes in math vary from country to country: not universal!

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Background for the study Observations gleaned from personal experience as co-teacher for ELLs in a mainstream mathematics classroom Instructional strategies Traditional lock-step method (all learners proceed at the same pace). Demonstration-mimicry-practice One size fits all strategy that did not include the language needs or experiences of ELLs

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Results No opportunity to practice arithmetic vocabulary. Does not foster additional language development or literacy skills. No opportunity to participate or co-construct the problem solving process. No opportunity to collaborate with classmates and create a community of learners. Compromises affective variables. No opportunity to discuss, justify, or explain the problem solving processes.

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Observation of additional content classrooms and instructional delivery led to similar results more presentation than interaction. Teacher attitudes regarding social interaction in the classroom were similar across content areas: specifically that it results in an ill-managed classroom and promotes cheating

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Research Purpose Deeper investigation revealed: Increase in number of ELLs Increase in the drop out rate for ELLs (Fry, 2008; Menken, 2008). ELL scores of high school exit exams in math 30-40% lower than non-ELLs (Center on Education Policy, 2005). Teacher preparation Paucity of research

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Social Interaction Why social interaction? Reviewed literature to develop a conceptual framework Sociocultural Approaches to Learning and Development

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Conceptual Framework Important characteristics of interaction Questioning and wait time are important tools: to engage students in thinking increase active participation provide students an opportunity to demonstrate problem solving processes and vocabulary under the guidance of a peer or more capable other.

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Study: Research Questions How do advanced ELL tutors interact with ELL tutees to develop ownership of mathematical problem solving activity? How do effective tutors interact differently from ineffective tutors? What types of tutor questions support active participation, independent problem solving, and accurate completion of problem sets as tutees engage in small group activities (ownership)? In what way does tutors wait time contribute to tutees use of arithmetic vocabulary?

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Study: Research Procedure 18 dyads 15 ELL (expert)student/ELL (novice) student 3 teacher/ELL (novice) student Five math problems PSSA grade 11 mathematics test item sampler Distributed to tutees to indicate difficulty in problem solution Problem sets were divided among tutors

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Thirty minutes for problem solving Dyads were allowed no more than 30 minutes to complete a sample set of 2-5 problems.

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Research: Analytic Methods Open line by line coding/selective coding Data from recordings were transcribed via longhand in conjunction with field notes. Questions were highlighted. Pauses were highlighted as possible wait time. Arithmetic vocabulary highlighted. Compared and contrasted data within and across dyads to identify patterns and themes that emerged by focusing on tutee response, questions, and use of wait time.

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Procedural questions: Provided the least amount of instructional support. Provided the least amount of opportunity for the tutee to actively participate or demonstrate independent problem solving. Consisted primarily of yes/no answers. Used by tutors as comprehension checks, prompts for acknowledgement, or agreement, and to direct attention to the problem solving process. Often posed after the introduction of an activity, explanation of procedures, or elicit tutee confidence in reproducing the activity. Often associated with moving the activity forward

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Guiding questions: Provide a high level of instructional support. Provided opportunity for tutee to actively participate in co-constructing the problem solving process. Consist of known answer/display questions. Used by tutors during the problem solving task to effectively model a problem solving process, assess background knowledge, elicit a response, or focus attention on an activity. Often encouraged remediation or tutor demonstration of proper techniques or steps to collaboratively arrive at a correct solution to the activity.

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Reflection questions: Provide a mid-range level of instructional support as guidance if the tutee requested assistance or arrived at an incorrect solution. High level of opportunity for tutee to demonstrate independent problem solving. Questions are open, implicit, or vague. Often used at the close of a task to determine internalization or comprehension of the problem solving activity. Offered by the tutor as a means for the tutee to physically demonstrate, explain, or justify the problem solving process, reflection questions offered tutors a means to assess what the tutee had learned.

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Pause between Questions/response sequences Response/reflection sequences totaled three seconds or more. Instances of wait time promoting arithmetic vocabulary Tina In each of the three dyadic interactions between Tina and a tutee, there was at least one episode of interaction in which wait time increased the use of arithmetic vocabulary. Jean In two of the dyadic interactions between Jean and the tutee, wait time contributed at least once in the tutees use of arithmetic vocabulary.

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Jean: most effective Interaction Style Introduces the lesson and gives the tutee a visual and verbal preview of what they will do in order to solve the problem Fractures the problem into manageable fragments. Uses a variety of question types (procedural and primarily guiding) to engage tutee in active participation of co-constructing the problem solving process as they solve each piece of the fractured problem to produce a unified problem set.

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Uses guiding questions and wait time to prompt tutees use of arithmetic vocabulary. Uses reflection questions to offer the tutee an opportunity to physically demonstrate independent problem solving of a similar problem and often accompanies this with a request to demonstrate or justify the process the tutee had completed. Tutee usually demonstrated ownership of the mathematic problem solving activity in as little as 5 minutes.

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Cindy: least effective Interaction Style Seldom offers a demonstration, explanation, or preview of the problem solving process necessary to complete the mathematic activity. Occasionally uses guiding questions to encourage active participation, however it is often unclear what the tutee is solving for.

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Often fractures the problem similar to Jean, however, waters down the content and vocabulary with simplified versions that the tutees would not encounter in a grade appropriate mathematics assessment. Does not use reflection questions, therefore it is difficult to determine if tutee ownership of the mathematics activity did occur.

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Key Benefits of the Study Evidence that interaction can be used in the secondary mathematics classroom (and across the curriculum)to develop ELL ownership of mathematics problem solving activity The importance of tutor questions and wait time The combination of the three types of questions, procedural, guiding, and reflection in conjunction with episodes of wait time contributed to developing tutee ownership of the mathematic activity.

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Key Benefits Continued Social interaction This provides evidence of the positive effects of social interaction in mathematics learning which dispels the myth that dyadic or small group work results in bad habits (cheating, copying) and provides no evidence of independent problem solving.

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Practical solutions for secondary mathematics teachers of ELLs Example: Visual representation of the problem accompanied by explanation of the vocabulary and/or problem solving process Poses a procedural question as a comprehension check or to direct attention to a particular area of the problem. Fractures the problem. Allows students to work in small groups and then actively engage (through guiding questions) in co- constructing the problem solving process as a whole class.

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Pose reflection questions to allow students an opportunity to reflect, debate, discuss one another's answers Pair students in dyads to work on a set of problems similar to the one previously demonstrated. Circulate room and offer extra help where needed Upon completion of the tutoring episode, teacher can revisit problem sets, fracture them, and once again actively engage whole class in reassembling one or two activities. Assign individual problems sets for independent practice.

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Implications Professional development for teachers and pre-service education Remember: it is not only about interaction, but the type of strategies we use to interact in the classroom:

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Conclusion It is extremely important that all teachers understand how to implement effective problem solving strategies for ELLs (and all students) through social interaction.

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References Fraivillig, J., Murphy, L. A., & Fuson, K. C. (1999) Advancing childrens mathematical thinking in Everyday Mathematics reform classrooms. Journal for Research in Mathematics Education 30(2): Gibbons, P. (2003). Mediating language learning: Teacher interactions with ESL students in a content- based classroom. TESOL Quarterly, 37 (2), Hufferd-Ackles, K., Fuson, K.C., & Sherin, M.G. (2004). Describing levels and components of a math- talking learning community. Journal for Research in Mathematics Education, 35(2), Khisty, L. L., & Chval, K. B. (2002). Pedagogic discourse and equity in mathematics when teachers talk matters. Mathematics Education Research Journal, 14, Kim. Y. (2010). Scaffolding through qustions in upper elementary ELL learning. Literacy Teaching and Learning Volume 15, (1&2) pp Lampert, M. (2001).Teaching Problems and the Problems with Teaching. New Haven, CT: Yale University Press Lantolf, J. & G. Appel (1998). Vygotskian Approaches to Second Language Research. Norwood, NJ: Ablex Publishing Corporation

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References Continued Lantolf, J.P. & Poehner, M.E. (2010). Dynamic assessment in the classroom: Vygotskian praxis for second language development. Language Teaching Research (15 ), Lantolf, J.P. & Poehner, M.E. (2004). Dynamic assessment of L2 development: bringing the past into the future. Journal of Applied Linguistics, 1(1), pp Lave, J. (1988). Cognition in practice: Mind, mathematics, and culture in everyday life. Cambridge, UK: Cambridge University Press. Lave, J., & Wenger, E. (1991) Situated learning: Legitimate peripheral participation. Cambridge: Cambridge University Press. Markee, N. P. P. (2004c). Zones of interactional transition in ESL classes. Modern Language Journal, 88, Menken, K. (2008). English learners left behind: Standardized testing as language policy. Clevedon, Avon: Multilingual Matters.

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References Continued Moschkovich, J. (1999). Supporting the participation of English language learners in mathematical discussions. For the Learning of Mathematics, 19, Moschkovich, J. (2002). A situated and sociocultural perspective on bilingual mathematics learners. Mathematical Thinking and Learning 4 (2&3): Moschkovich, J. (2007). Examining mathematical discourse practices. For the Learning of Mathematics, 27, Parks, N.A.(2010). Explict versus implicit questioning: Inviting all children to thin mathematically. Teachers College Record. Volume 112 (7) pp Rowe, M.B. (1986). Wait time: Slowing down may be a way of speeding up! Journal of Teacher Education 37, pp Rowe, M.B. (1974).Wait-Time and rewards as instructional variables, Their influence on language, logic, and fate Control: Part one-wait-time." Journal of Research in Science Teaching 11, 2:

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References Continued Saxe, G. B.(1996) Studying Cognitive Development In Sociocultural Context. In D. Jessor, A. Colby, & R. Shewder (Eds.) Ethnographic approaches to the study of human development (pp ). Chicago: University of Chicago Press, Saxe, G. B. (1991). Culture and cognitive development: Studies in mathematical understandings. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc. Scribner, S. (1984). Studying working intelligence. In B. Rogoff & J. Lave (Eds.), Everyday Cognition: Its development in social context (pp. 9-40). Cambridge, MA: Harvard University Press. Stahl, R. (1995). Using think tine and wait time skillfully in the class. Eric Digest: ERIC Identifier: ED Storch, N. (2004). Using activity theory to explain differences in patterns of dyadic interactions in an ESL class. The Canadian Modern Language Review, 60(4), Swain, M. (2000). The output hypothesis and beyond: Mediating acquisition through collaborative dialogue. In J. Lantolf (Ed.), Sociocultural theory and second language learning (pp ). Oxford: Oxford University Press.

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References Continued Swain, M., & Lapkin, S. (1998). Interaction and second language learning: Two adolescent French immersion students working together. Modern Language Journal, 82, Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, New York: Cambridge University Press. Yackel, E., Cobb, P., & Wood, T. (1991). Small-group interactionsa as a source of learning opportunities in second-grade mathematics. Journal for Research in Mathematics Education, 22,

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