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Responding to Children’s Thinking and Diversity: A Reflection on 20 years of Research

Responding to Children’s Thinking and Diversity: A Reflection on 20 years of Research. Megan Loef Franke UCLA. Cognitively Guided Instruction. Thomas Carpenter (University of Wisconsin) Elizabeth Fennema (University of Wisconsin) Linda Levi (University of Wisconsin)

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Responding to Children’s Thinking and Diversity: A Reflection on 20 years of Research

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  1. Responding to Children’s Thinking and Diversity: A Reflection on 20 years of Research Megan Loef Franke UCLA

  2. Cognitively Guided Instruction • Thomas Carpenter (University of Wisconsin) • Elizabeth Fennema (University of Wisconsin) • Linda Levi (University of Wisconsin) • Susan Empson (University of Texas) • Ellen Ansell (University of Pittsburgh) • Vicki Jacobs (San Diego State University) • Elham Kazemi (University of Washington) • Dan Battey (Arizona State Univ) • Annie, Mazie, Sue, Barb, Lilliam, Jo Ann, Kim, Janet, and many, many other teachers

  3. Presentation Overview • Why focus on Children’s Mathematical Thinking • Adding it Up • Equity • Making use of the development of children’s mathematical thinking • Research findings • Supporting the development of children’s mathematical thinking in classrooms

  4. Considering Understanding: Adding it Up • Recognizing that no term captures completely all aspects of expertise, competence, knowledge, and facility in mathematics, we have chosen mathematical proficiency • five interwoven, interdependent strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, productive disposition. • Mathematical proficiency is not a one-dimensional trait, and it cannot be achieved by focusing on just one or two of these strands

  5. About children’s thinking • Children come to school with mathematical knowledge • Children’s knowledge develops through well documented trajectories • Development of children’s thinking quite robust • Development of children’s thinking does not match the way adults solve problems

  6. Development of Children’s Mathematical Thinking Janelle has 7 trolls in her collection. How many more trolls will Janelle need to buy to have 11 trolls altogether? How do you think children will solve this problem? Watch what children can do… Direct modeling Counting strategy Derived Fact Recall

  7. How can a focus on children’s thinking help? • Notice what students’ can do • Make decisions based on what students’ know • Press for understanding

  8. How can a focus on children’s thinking help? • Create multiple ways to participate • Support the development of mathematical identity—not just one way, sense making, question asking…

  9. CGI Research and Development ExperimentalStudy LongitudinalStudy School Case Study Teacher Case Studies Follow up Teacher/School ExperimentalStudy Development of communities of inquiry Development of tools to support learning in practice Learning about the development of students’ mathematical thinking in classrooms First Grade +/- K- 3 +/- x/÷ p.v. All school +/- x/÷ p.v. K-5 Algebraic thinking

  10. Evidence that Attending to Student Thinking Can Make a Difference • CGI provides evidence that teachers’ classroom practice that • includes eliciting and making public student thinking, • involves eliciting multiple strategies, • focuses on solving word problems and • uses what is heard from students to make instructional decisions leads to the development of student understanding

  11. Evidence that Attending to Student Thinking Can Make a Difference • Teachers who drew on • detailed knowledge of the development of students’ mathematical thinking within a domain • an organization of student thinking in relation to the mathematical content • notions that they could continue to learn from their practice …identity supported the development of student understanding

  12. Supporting teachers to make use of students’ mathematical thinking • There is no single pattern or trajectory for teachers as they come to make use of children’s thinking • Can get teachers to ask students how they solved problems • Challenging to support teachers to make use of what they hear from students, to engage students in comparing strategies, to move forward in their trajectories

  13. Moving forward…learning more to support teacher learning and practice • Pushing on the research • Learning through professional development

  14. Moving towards understanding the details of practice through research • Listening to students talk makes it possible for the teachers (and other students) to monitor students’ mathematical thinking • The act of talking can itself help students develop improved understanding • Explaining to other students is positively related to achievement outcomes, even when controlling for prior achievement • (Brown & Palincsar, 1989; Fuchs, Fuchs, Hamlett, Phillips, Karns, & Dutka, 1997; King, 1992; Nattiv, 1994; Peterson, Janicki, & Swing, 1981; Saxe, Gearhart, Note, & Paduano, 1993; Slavin, 1987; Webb, 1991; Yackel, Cobb, Wood, Wheatley, & Merkel, 1990). • Less is known about teacher practices that are most effective for producing high-level discourse in the classroom

  15. Details: Supporting the development of students’ mathematical thinking In classrooms where: Students gave correct and complete explanations Students scored the highest on the assessments Teachers: • Used a fairly coherent set of problems • Asked questions very specific to what students said • Engaged students in thinking and talking about important mathematical ideas arising out of their suggestions • All students participated in conversations about the mathematics

  16. Learning through professional development • develop relationships: create a community where teachers can learn together about the teaching and learning of mathematics • where the activities of the community were embedded in teachers’ everyday work • make space for teachers to share their histories and make their practice public • focus on the details and structures around students’ mathematical thinking • Focus on what students can do (Counter-storytelling) • attention to the artifacts and language that support the development of students’ mathematical thinking in practice

  17. Framework for the development of student strategies within mathematical domains • Problem types • Video of students and classrooms • Language How did you get that? Does that always work? Strategy and problem names Number sentence index cards Join Change Unknown Avita has 7 rocks. How many more rocks does she need to collect to have 11 rocks altogether? Join Result Unknown … Artifacts in our Professional Development work…

  18. Artifact Travel • Ongoing use across settings • Attention to and unpacking of classroom use in PD • Trace where we are with ideas around artifact • Helps to see teacher use of artifact in both PD and classrooms – raise questions, note inconsistencies, conflict etc..

  19. How artifacts support learning and practice • Focused on creating and negotiating meaning • Focused conversation across and within communities of practice • Supported the development of language and interaction that could be used to support the development of new relationships • Supported story telling across boundaries and be used to develop counter stories • Purposeful • Challenged the existing cultural practices

  20. Development of Children’s Mathematical Thinking Tom has 102 dog biscuits. His Dog Harmony eats 12 biscuits a day. How many days will it take Harmony to eat all of the dog biscuits? Let’s see what children can do…

  21. Attending to the Details of Children’s Mathematical Thinking Allows Teachers to: • Notice what students can do • Make decisions that build on what students know • Create openings for varied participation • Support the development of students who think of themselves as capable of making sense of mathematics

  22. Development of Children’s Mathematical Thinking Lucy had 38 dollars. One weekend she earned 25 making dollars raking leaves for her neighbors. How much money did Lucy have then? Watch what children can do… What can you do? Count by tens, solve problems using 20 and 30, take numbers apart and put them back together.

  23. Mathematical Proficiency • conceptual understanding—comprehension of mathematical concepts, operations, and relations • procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately • strategic competence—ability to formulate, represent, and solve mathematical problems • adaptive reasoning—capacity for logical thought, reflection, explanation, and justification • productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

  24. Development of Children’s Mathematical Thinking 8 + 4 =  + 5 Can students solve without computing each side? Let’s watch David…

  25. Equal Sign Data (8+4=  +5) 1Falkner, K., Levi, L., & Carpenter, T. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6, 232-6.

  26. Evidence that Attending to Student Thinking Can Make a Difference • Students constantly surprise us… • Kindergarten data • Fractions • Algebraic thinking

  27. Focusing on Making Student Thinking Explicit • need to be able to use student strategies as the center of the workgroup conversation, as one of the tools teachers interact with • expertise shared • change in power structures, teacher as expert • provides an explicit trace of the group’s thinking • extends to other communities of practice • centers the role of the professional developer

  28. Moving Towards the Details of Practice • Need to know more about student participation in mathematics classrooms if we are to support teaching • Often large scale studies focus on what occurs in public discourse • Smaller scale studies document more specifically student participation and what that means for student learning Forman, et al, 1998; Lampert, 2001; Moschkovich, 2002; O’Connor & Michaels, 1996; Palincsar & Brown, 1984, 1989; Yackel, Cobb, & Wood, 1991 • Want to look to the relationship between student participation, teaching, the mathematics and student outcomes

  29. Development of Children’s Mathematical Thinking 19 Children are taking a mini-bus to the zoo. They will have to sit either 2 or 3 to a seat. The bus has 7 seats. How many children will have to sit 3 to a seat and how many can sit 2 to a seat? How will children solve it? How about a kindergartener? 59% had a correct strategy 51% correct answer, 33% 1st graders, 26% 2nd graders

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