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  1. CRESMET Project Pathways: Overview and Evolution Marilyn P. Carlson Project Pathways PI Director, CRESMET Professor, Department of Mathematics and Statistics Arizona State University This work was supported, in part, by grant no. 9876127 from the National Science Foundation

  2. Project Pathways • Partnership of ASU and five school districts • Primary Goal: • To produce a research-developed, refined & tested model of inservice professional development for secondary mathematics and science teachers • Core Strategies: • Four integrated math/science graduate courses + linked teacher professional learning communities (lesson study approach)

  3. Local Conditions • Students • Arizona has the nation’s highest school dropout rates • Fewer than 25% of Arizona students scored “proficient” or higher on the 2003 National Assessment of Educational Progress of mathematics and science • Teachers • 58% of Arizona teachers do not hold a degree in the subject that they teach • Arizona teachers receive professional development that has been categorized as scattershot, seldom, and shallow • Spent an average two days per year in professional development activities • Statistics from NCES, 2003, Killion, 2002, Sowell, 1995 .

  4. Pathways Objectives for Teachers • Deepen teachers’ understanding of foundational mathematics & science concepts and their connections • Understanding and use of covariational reasoning and function as connecting themes • Improve teachers’ reasoning abilities and STEM habits of mind (problem solving, scientific inquiry, engineering design) • Support teachers in adopting “expert” beliefs about STEM learning, STEM teaching, and STEM methods (See STEM BILT Taxonomy) • Focus on promoting content knowledge for teaching, e.g., the processes and complexities of acquiring understanding of key ideas. • Support teachers in reflecting on and modifying their classroom instruction

  5. Intervention • Courses • Focus on connections and coherence • Focus on covariational reasoning and function as connecting themes • Focus on developing STEM habits • Model student centered instruction

  6. Overview • Research that supports this focus: • Conceptual frameworks (based on multiple years of qualitative studies) have informed the emergence of a Function Inventory (Thompson, 1994; Carlson, 1998; Carlson, Jacobs, Coe, Larsen, Hsu, 2000; Oehrtman, 2004) • Qualitative research and problem solving framework (Carlson, 1998; Carlson & Bloom, 2005) have informed the emergence of a STEM ‘habits of mind’ framework • A characterization of effective problem solving behaviors

  7. Revise Cognitive Frameworks The Reflexive Relationship Between Frameworks and Intervention • One Example Carlson & Bloom Problem Solving Framework -Describes effective mathematical practices Conceptual Frameworks (e.g., Covariation, FTC, Function) -Characterizes understanding (e.g., reasoning abilities, connections, notational issues) Theoretical grounding for -Designing curricular modules -Determining course structure -Instrument development Lens for researching emerging practices and understandings

  8. Instruments for Assessing Pathways Progress and Effectiveness • Function Concept Inventory • 25 Item multiple choice instruments that has been validated over the past 7 years • Beliefs about STEM habits and STEM teaching • Developed--early in the validation process • PLC Observation Protocol • Still collecting qualitative data--not yet developed • RTOP • Already validated and published

  9. The Bottle ProblemImagine this bottle filling with water. Sketch a graph of the height as a function of the amount of water that’s in the bottle

  10. Mental Actions of the Covariational Reasoning Framework • MA1) Coordinating one variable with changes in the other variable • MA2) Coordinating the direction of change in one variable with changes in the other variable (e.g., increasing, decreasing) • MA3) Coordinating the amount of change of one variable with changes in the other variable • MA4) Coodinating the average rate of change of one variable (with respect to the other variable) with uniform changes in the other variable • MA5) Coordinating the instantaneous rate of change of one variable (with respect to the other variable) with continuous changes in the other variable • (Carlson, Jacobs, Coe, Larsen, Hsu, 2002)

  11. Covariational Reasoning: A Foundational Reasoning Ability for Understanding and Using Big Ideas of CalculusDerivativeAccumulation

  12. Problem Solving Process • (Carlson & Bloom, 2005)

  13. Major Findings that Influenced Intervention Adaptations: • Pace of course, amount and nature of homework, and expectations for classroom participation influenced the level of conceptual learning and exhibition of problem solving behaviors for STEM teachers. • Multiple viewing and coding of classroom videos revealed • Development of a deep understanding of fundamental concepts such proportionality, rate-of-change and exponential growth is complex • Pacing, homework and enactment of classroom practices were issues • Adjustments were made to modules and homework

  14. Select Findings: • Analysis of qualitative data revealed irregularities in teachers’ enactment of problem solving behaviors during class • Some teachers were not engaged in making sense of the tasks/problems during the classroom activities • Teachers sometimes pretended to understand when they did not • Teachers communication relative to the mathematics was often incoherent

  15. In the third cohort explicit “rules of engagement” were negotiated • Explicit attention to enacting the Rules of Engagement during interaction or discourse • Speaking with meaning • Mathematical Integrity • Sense Making • Respecting the learning process of colleagues

  16. Speaking with Meaning • Speaking with meaning implies that responses are conceptually based, conjectures are based on logic, conclusions are supported by a mathematical argument, and explanations are given using the quantities involved. • Examples: Contrast responses to the bottle problem.

  17. Early results from cohort 3 are encouraging: • Shifts in teachers’ understanding of concepts was significant (e.g., PCA mean score shifted from 17 to 23) • The “rules of engagement” became spontaneous • Teachers reported that attention to speaking with meaning had impacted: • Their attention to student thinking, their communication patterns with students, their understanding of key ideas, etc. • Positive shifts were noted on STEM Beliefs instrument

  18. Early results revealed perceived ‘factors of resistance” for shifting instruction towards coherent and conceptually focused lessons. • Textbook does not support teaching ideas or concepts • Standardized tests do not value understanding • Students are not smart enough to work thought provoking tasks • Teachers’ images of teaching involves “stand and deliver” instruction, procedural learning, memorization

  19. Pathways Response: A Professional Learning Community (PLC) • A Pathways Professional Learning Community • 3-7 teachers who meet for 1-2 hours weekly to reflect on and discuss what is involved in knowing, learning and teaching concepts. (A school based facilitator is responsible for focusing the discussions.) • Activities of the PLC: • Unpack and discuss what is involved in understanding an idea • Discuss and evaluate student thinking (interview students) • Reflect on the effectiveness of instruction in promoting student learning • Eventually move to cycle of planning, developing, teaching, studying and refining conceptually focused lessons

  20. PLC Intervention • Facilitators have received training to manage discourse among PLC members • Facilitators manage agenda developed by Pathways faculty • Explicit attention to enacting the “Rules of Engagement” during interaction or discourse • Speaking with meaning • Mathematical Integrity • Sense Making • Respecting the learning process of colleagues

  21. Research Questions • What are the attributes of a “high functioning” PLC? • What variables promote “quality discourse” about knowing, learning, and teaching mathematics?

  22. Methods: Grounded Theory • Videotaped, reviewed and coded videos of four PLCs • Teams of 2 RAs were responsible for coding each PLC throughout the semester. Initially coded for enactment of (and missed opportunities to enact) the rules of engagement • Documented emerging patterns • Share results weekly and collaborate to refine our original definitions

  23. What We Are Learning • The facilitator is a critical variable in determining the quality of the PLC discourse • Facilitators varied relative to their: • Conceptual knowledge • Level of inquiry about PLC members’ thinking • Level of inquiry in regard to their own teaching • Commitment to the goals of the PLC • Ability to demand speaking with meaning among PLC members • Observable differences in PLC discourse relative to: Decentering

  24. Facilitator Decentering • Definition • adopting a perspective that is not one’s own • Involves the ways a person adjusts his or her behavior in order to influence another in specific ways

  25. Manifestations of Decentering • Facilitator makes no attempt to build a model of other member’s thinking • Facilitator explains and moves on • Facilitator appears to build a partial model of a member’s thinking • Facilitator explains; then asks questions to determine if a PLC member understood some aspects of her/his explanation • Facilitator appears to build a model of a member’s thinking, but does not use that model usefully in communication • Facilitator listens and/or asks PLC member questions and provides statements that indicate that he/she understands PLC member’s thinking, but facilitator reacts by explain her/his thinking again

  26. Manifestations of decentering • Facilitator builds a model of a member’s thinking for the purpose of moving he/she to her/his way of thinking • Facilitator listens to a member’s thinking and recognizes that it is different than her/his own; then acts in ways to move this person to her/his way of thinking • Facilitator builds a model of a member’s thinking and acts in ways to understand the rationality of this member’s thinking • Facilitator questions a member to understand her/his thinking. Also, facilitator create a new model about how this member might understand his/her way of thinking.Facilitator then adjusts her/his interactions (questions, drawings, statements, etc.) to take into account the member’s thinking and how this member might understand him/her.

  27. Summary Results • Five manifestations of decentering were observed • Facilitators who made efforts to understand the thinking and perspective of other PLC members (decentering) were better able to engage the members of the community in meaningful discourse • Facilitator’s level of understanding of the concept that was central to the lesson influenced the quality of the PLC discourse • Revealed by her/his questions, choice of tasks, and choice to pursue an exchange with a specific PLC member

  28. Implications and Further Analysis • Continue to explore attributes of a facilitator that lead to “high quality” discourse among PLC members • Follow PLC members into classroom to ascertain relationships between PLC behavior and classroom behavior • Use new knowledge • Selection of facilitators • Improve PLC facilitator training • Theoretical framing for the Learning Community Observation Protocol

  29. Acknowledgements • The research reported in this paper was supported by the National Science Foundation under grant number: HER-0412537. All opinions expressed are solely those of the authors