Report

La Cosecha 2012 Santa Fe, New Mexico DEVELOPING BI-LITERACY THROUGH MATHEMATICS IN KINDERGARTEN CLASSROOMS Sylvia Celedón-Pattichis, Ph.D. University of New Mexico National Science Foundation Award No. ESI-0424983 OVERVIEW OF THE SESSION Center for the Mathematics Education of Latinas/os (CEMELA) Background of Kindergarten Study Cognitively Guided Instruction (CGI) Activities and video clips that engage participants in understanding children’s mathematical thinking and ways to support discursive habits such as listening, speaking, reading, writing, and representing solutions. BACKGROUND OF CEMELA AND KINDERGARTEN STUDY OVERVIEW OF THE SESSION Center for the Mathematics Education of Latinas/os (CEMELA) Background on young children and Latina/o students Cognitively Guided Instruction (CGI) Professional development work with bilingual kindergarten teachers Activities and video clips that engage participants in understanding children’s mathematical thinking and ways to support discursive habits such as listening, speaking, reading, writing, and representing solutions. CEMELA University of Arizona: Marta Civil, Virginia Horak, and Luis Moll University of California at Santa Cruz: Judit Moschkovich and Kip Téllez University of Illinois at Chicago: Lena Khisty and Aria Razfar University of New Mexico: Rick Kitchen and Sylvia Celedón-Pattichis CEMELA GOALS To develop an integrated knowledge model that connects mathematics teaching and learning to the cultural, social, political, and linguistic context of Latina/o children and To increase the number of mathematics educators with this integrated knowledge to ultimately improve the mathematics education of low-income Latina/o children. CONTEXT: THE SCHOOL An urban elementary bilingual school in New Mexico Promotes bilingualism and biliteracy Reform-based mathematics curriculum Mathematics taught in Spanish 86% Latina/o student population (mostly Mexican immigrants) 100% free or reduced meals CONTEXT: THE TEACHERS Larger study included 7 teachers. The focus is on two teachers. Ms. Arenas--Kinder--Experienced-Guatemala Ms. Carrera--Kinder--Novice--Mexico Both teachers had attended CEMELA Summer Institutes, had in-class support, and participated in 3 workshops/semester PORTRAIT OF INSTRUCTION Problem solving lessons conducted twice a week, for about 30 minutes Average of 3 problems per lesson Both whole group and small group formats used Students had access to a range of tools TEACHING MATHEMATICS TO EMERGENT BILINGUAL (EB) STUDENTS To foster mathematics academic literacy in the bilingual classroom, we need: High expectations for students’ academic achievement and maintaining the native language (while developing English) Understanding language as a resource instead of a deficiency Fostering EB participation in mathematics conversations besides vocabulary (Moschkovich, 2007, 2010) COGNITIVELY GUIDED INSTRUCTION (CGI) (Carpenter et al., 1999) Predict the percentage of kindergartners who can solve the different type of word problems by the end of the school year. WHAT THIS RESEARCH There TELLS US… is often an underestimated problem solving capacity of young children (CGI Studies, Carpenter et al., 1999). Young children can engage in problem solving BEFORE they master basic mathematics facts. COGNITIVELY GUIDED INSTRUCTION (CGI) AS A FRAMEWORK (CARPENTER ET AL.,1999) It is a framework for understanding children’s mathematical thinking. Children enter school with a great deal of informal and intuitive knowledge of mathematics. Bridging students’ experiential knowledge with formal school mathematics is critical. Use of context-rich word problems is based on knowledge of students’ communities and the mathematical practices in which their families engage (González, Moll, & Amanti, 2005). PROBLEM TYPES Join (Result Unknown) Connie had 5 marbles. Juan gave her 8 more. How many does Connie have altogether? (Change unknown) Connie has 5 marbles. How many does she need to have 13 altogether? (Start Unknown) Connie had some marbles. Juan gave her 5 more. Now she has 13. How many marbles did Connie have to start with? Separate (Result Unknown) Connie had 13 marbles. She gave 5 to Juan. How many marbles does Connie have left? (Change unknown) Connie had 13 marbles. She gave some to Juan. Now she has 5 marbles left. How many marbles did Connie give to Juan? (Start Unknown) Connie had some marbles. She gave 5 to Juan. Now she has 8 marbles left. How many marbles did Connie have to start with? Part-Part Whole (Whole Unknown) Connie has 5 red marbles and 8 blue marbles. How many marbles does she have? Compare (Difference Unknown) Connie has 13 marbles. Juan has 5 marbles. How many more does Connie have than Juan? (Part Unknown) Connie has 13 marbles. 8 are blue. How many red marbles does Connie have? (Compare Quantity Unknown) Juan has 5 marbles. Connie has 8 more. How many does she have? (Referent Unknown) Connie has 13 marbles. She has 5 more than Juan. How many marbles does Juan have? TYPES OF WORD PROBLEMS Kinder Mastered in 2nd WHAT WE LEARNED FROM THE KINDERGARTEN STUDY ABOUT SUPPORING DISCURSIVE MATHEMATICAL HABITS (Sfard, 2000, 2001) 1. MATHEMATICS ACADEMIC LITERACY: CONTEXTUALIZING THROUGH STORYTELLING Introducing problem solving through “storytelling” conversations. Problems that reflect familiar contexts invite students to draw upon lived experiences to make sense of mathematical ideas. The narrative structure of the problems scaffolds students’ explanations. Problems in the form of stories help students learn to represent mathematical ideas and connect multiple representations (e.g., drawings, symbols, objects). PRACTICE #1: VIDEO CASE Ms. Arenas’ Class, April of 2006 Typically began problem solving session with “Fíjense amorcitos pues, les voy a contar una historia” Video clip illustrates conversation that prompted division problem (9÷3) VIDEO CASE What opportunities are students afforded to represent their mathematical thinking (i.e., by listening, speaking, reading, and writing) in your own classrooms? 2. MATHEMATICS ACADEMIC LITERACY: MULTIMODAL REPRESENTATION Fostering the use of multimodal approaches (i.e., pictorial, symbolic, and written) to communicate the mathematical thinking. Representing information in nonlinguistic ways is also an important consideration for mathematics academic literacy development. Representing problem solving strategies in different ways: explaining, direct modeling, drawing, number sentence Video of students solving a subtraction problem (Separate Result Unknown) MULTIPLICATION PROBLEM Students learn to represent solutions pictorially. MATHEMATICS ACADEMIC LITERACY: DEVELOPING AND COMMUNICATING EFFECTIVE PROBLEM SOLVING STRATEGIES Type Students develop more effective and sophisticated problem High-frequency solving strategies by vocabulary listening and using oral /phrases and written explanations to explain their General mathematical thinking. vocabulary Students must draw /phrases from all vocabulary types participating in Specialized mathematical vocabulary conversations. /phrases Spanish English Mas mismo Hay More Same There is/are Igual Contar Es Mayor que Es Menor que Equal Count Is More than Is Less than Sumar Restar Contar de cinco en cinco Oración numérica Add Subtract Count by fives Numeric sentence 3. MATHEMATICS ACADEMIC LITERACY: DEVELOPING AND COMMUNICATING EFFICIENT PROBLEM SOLVING STRATEGIES Video: Multiplication Comparing strategies Which strategy is more efficient? I had three boxes. In each box I had five lollipops. How many lollipops did I have? 3. MATHEMATICS ACADEMIC LITERACY: DEVELOPING AND COMMUNICATING EFFICIENT PROBLEM SOLVING STRATEGIES Consistently, teachers used scaffolding strategies and provided multiple opportunities to engage in mathematics conversations. Modeling mathematical ways of talking Revoicing students’ explanations: “So you’re saying that he counted by fives?” “Oh, so you are saying that you counted on, you started at 4 and then counted on, 5, 6, 7, 8.)” 3. MATHEMATICS ACADEMIC LITERACY: DEVELOPING AND COMMUNICATING EFFICIENT PROBLEM SOLVING STRATEGIES Teachers used questioning to: Make sense of the problem and search for a solution How would you describe the problem in your own words? How would you describe what you are trying to find? What do you notice about...? What information is given in the problem? Construct arguments and explain reasoning Would you explain to me how you figured this out? How did you count? Which way to solve the problem is faster? Why? How can we be sure that...? / How could you prove that...? 3. MATHEMATICS ACADEMIC LITERACY: DEVELOPING AND COMMUNICATING EFFECTIVE PROBLEM SOLVING STRATEGIES Validating and generalizing mathematics procedures. “We were already generalizing and they were abbreviating the long addition process into multiplication process. (…) We try to validate and generalize algorithms in my class(…) we generalize and validate procedures. I let the students use the procedure that they feel better with, as soon as it is valid. You might ask me what happens when the algorithms are not valid? It doesn’t matter. It is good learning for them. When the algorithms are not valid, we try them with the whole class or the whole group, sometimes I do it with small groups, and then we validate it once and they learn by their errors and they’ll never use that algorithm again. And the advantage is that they have several different algorithms to use, so when they have to solve it quickly (…) they go with the one that is easier to them, and then they can perform more accurate and more quickly.” What are ways to support students in making a shift from problem solving to problem posing? 4. MATHEMATICS ACADEMIC LITERACY: FROM PROBLEM SOLVERS TO PROBLEM POSERS First, students develop and use their own strategies to solve problems: Direct modeling (using concrete objects or manipulatives) Counting strategies (counting up or down, counting on from, etc.) Finally, students construct their own “story” problem becoming “problem posers.” Students develop ownership and confidence as mathematics learners. MATH JOURNALS PROMOTE ACADEMIC LITERACY DEVELOPMENT Thinking: Requires in depth understanding of the structure of a problem Drawing: Representing the problem and solution Writing: Providing the story line for their own number stories Creating their own problems, students learn to pose problems in writing, representing the solution pictorially and symbolically. PARTITIVE DIVISION PROBLEMS Yo tenía ocho galletas y les di a mis amigas cuatro. The drawing shows how she distributed four cookies to each friend. The algorithm shows that she didn’t have any left. Clear understanding of what zero represents. DEVELOPING MATHEMATICS ACADEMIC LITERACY AND BEING PART OF A MATHEMATICS DISCOURSE COMMUNITY (CELEDÓN-PATTICHIS & TURNER, 2012). Students were afforded opportunities to hear and use the language needed for learning mathematics, necessary for appropriation (Chval & Khisty, 2009). Students progressively incorporated more accurate ways of explaining their ideas and strategies. ACKNOWLEDGEMENTS Dr. Erin Turner Dr. Sandra Musanti Dr. Mary Marshall UNM CEMELA Research Team Kindergarten Teachers CONTACT INFORMATION CEMELA Website: http://cemela.math.arizona.edu Sylvia Celedón-Pattichis [email protected] REFERENCES Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. Chval, K. B. & Khisty, L. L. (2009). Bilingual Latino students, writing and mathematics: A case study of successful teaching and learning. In R. Barwell (Ed.), Multilingualism in mathematics classrooms: Global perspectives (pp. 128-144). Bristol, UK: Multilingual Matters. Celedón-Pattichis, S. & Ramirez, N. (2012). Beyong good teaching: Advancing mathematics education for ELLs. Reston, VA: National Council of Teachers of Mathematics. Cummins, J. (2001). Empowering minority students: A framework for intervention. Harvard Educational Review, 71(4), 649-675. Cummins, J. (2005). Teaching the language of academic success: A framework for school-based language policies. In C. Leyba (Ed.),Schooling and language minority students: A theoretico-practical framework (3rd ed.) (pp. 3–32). Los Angeles: Legal Books Distributing. González, N., Moll, L. & Amanti, C. (2005). Funds of Knowledge: Theorizing Practices in Households, Communities, and Classrooms. Mahwah, NJ: Lawrence Earlbaum. Lea, M. & Street, B. (2006). The “academic literacies” model: Theory and applications. Theory into Practice, 45(4), 368–377. Moschkovich, J. N. (2010). Language and mathematics education: Multiple perspectives and directions for research. Charlotte: Information Age Publishing. Moschkowich, J. N. (2000). Learning mathematics in two languages: Moving from obstacles to resources. In W.G. Secada (Ed.), Changing the faces of mathematics: Perspectives on multicultural and gender equity (pp. 85–93). Reston, VA: National Council of Teachers of Mathematics. Musanti, S. I., Celedón-Pattichis, S., & Marshall, M. E. (2009). Reflections on language and mathematics problem solving: A case study of a bilingual first grade teacher. Bilingual Research Journal, 32(1), 25-41. Slavit, E. & Ernst-Slavit, G. (2007). Teaching mathematics and English to English Language Learners simultaneously. Middle School Journal, 39(2), 4-11 Turner, E., & Celedón-Pattichis, S. (2011). Problem solving and mathematical discourse among Latino/a kindergarten students: An analysis of opportunities to learn. Journal of Latinos and Education, 10(2), 1-24.