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

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)

Local Conditions
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
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 .

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

Focus on connections and coherence
Focus on covariational reasoning and function as connecting themes
Focus on developing STEM habits
Model student centered instruction

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

The Reflexive Relationship Between Frameworks and Intervention
One Example

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
Already validated and published

The Bottle Problem

Imagine this bottle filling with water. Sketch
 a graph of the height as a function of the
amount of water that’s in the bottle
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)

Covariational Reasoning:
A Foundational Reasoning Ability for Understanding and Using Big Ideas of Calculus

Derivative Accumulation
"Problem Solving Process"
Problem Solving Process
(Carlson & Bloom, 2005)

Slide 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

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

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

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.

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

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

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

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

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

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

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

Facilitator Decentering
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

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

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.

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

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

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