The Challenges of Scale: Designing
Learning Organizations for Instructional
Improvement in Mathematics
Paul Cobb
Vanderbilt University

Purpose
• Illustrate a way of conducting research studies that aim to inform the ongoing improvement of mathematics teaching and learning at scale

History of Failure
• The closer that an instructional innovation gets to what takes place between teachers and students in classrooms, the less likely it is that it will implemented and sustained on a large scale

Limited Impact of Research on
Classroom Practice
• Supporting students’ learning of central mathematical ideas
• Instructional materials
• Teachers’ instructional practices
• Supporting mathematics teachers’ development of high-quality instructional practices

Large-Scale Implementation
Projects
• Focus is primarily on teacher professional development
• Unanticipated “obstacles”
• Conflicts with other district initiatives
• Lack of understanding and/or support by school and district administrators

Large-Scale Implementation
• Flying blind: Little knowledge of the schools and districts in which they are working
• Reactive: Plans changed in response to
unanticipated obstacles
• Proactive: Anticipate school and district structures that might support mathematics teachers’ ongoing improvement of their instructional practices

Map Backwards From the Classroom
• Research on high-quality mathematics instruction
• Demands on the teacher
• Challenges of supporting the development of high-quality instructional practices
• School and district support structures

High-Quality Mathematics Instruction
•

Measuring With a Ten Bar

Measuring With a Ten Bar
• Edward: I think it ’ s 33 [points to where they have
marked 23 with the three cubes] because 10
[iterates the smurf bar once], 20 [iterates the smurf
bar a second time], 21, 22, 23 [counts the first,
second and third cubes within the second iteration]

Measuring With a Ten Bar
• Edward: Ten [iterates the smurf bar once], 20
[iterates the smurf bar again]. I change my mind.
She's right.
• T: What do you mean?
• Edward: This would be 20 [points to the end of
the second iteration].

Measuring With a Ten Bar
• T: What would be 20?
• Edward: This is 20 right here [places one hand at the beginning of the “ plank ” and the other at the
end of the second iteration]. This is the 20. Then, if I move it up just 3 more. There [breaks the bar to show 3 cubes and places the 3 cubes beyond
20]. That ’ s 23.

Measuring With a Ten Bar
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Classroom Discourse
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Demands on the Teacher
• Deep understanding of mathematics
• Mathematical knowledge for teaching
• Knowledge of how students’ reasoning develops in particular mathematical domains
• Know-in-practice how to pursuing a mathematical agenda by building on students’
(diverse) contributions

Improvement in Instructional Practices
• Students have to adjust to the teacher
• Teaching a routine activity
• Covering instructional objectives + classroom management
• Teacher adjusts instruction to the students
• Ongoing assessment of student reasoning
• Non-routine -- a complex and demanding activity

Framing Instructional Improvement at
Scale as a Research Issue
• Series of conjectures about school and district structures that might support teachers’ ongoing learning
• Instruments to document the institutional setting of mathematics teaching
• Extent to which the conjectured support structures have been established

Research Plan
• Four urban districts
• High proportion of students from traditionally underserved groups of students
• Limited financial resources
• Most districts clueless about how to respond productively to high-stakes accountability
• A small minority have reasonably worked out strategies

Research Plan
• Document district plans for improving middle-
school mathematics
• 6-10 middle schools - 30 teachers
• Four rounds of yearly data collection
• First year: Baseline data
• Document change over a three-year period in each district

Data Collection
• Institutional setting of mathematics teaching
• Audio-recorded interviews and surveys
• Quality of teacher professional development
• Video-recordings
• Quality of instructional materials and resources
• Artifact collection
• Quality of teachers’ instructional practices
• Video-recordings of two consecutive classroom lessons
• Teachers’ mathematical knowledge for teaching
• Student mathematics achievement data

Add Value to Districts’ Improvement
Efforts
• Feed back results of analyses to districts
• Gap analysis -- how district’s plan is actually playing out in schools
• Recommend actionable adjustments that might make each district’s improvement design more effective
• Design experiment at the level of the district

Research Team
Paul Cobb
Erin Henrick
Chuck Munter
Tom Smith
Kara Jackson
Sarah Green
John Murphy
Lynsey Gibbons
Karin Katterfeld
Glenn Colby
Annie Garrison

One District as an Illustrative Case
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Conjecture: Teacher Networks
• US teachers typically work in isolation
• Social support from colleagues in developing demanding instructional practices
• Focus of teacher interactions
• Classroom instructional practice
• Depth of teacher interactions
• Mathematical intent of instructional tasks
• Student reasoning strategies

Conjecture: Key Resources for
Teacher Networks
• Time built into the school schedule for collaboration among mathematics teachers
• Access to colleagues who have already developed relatively sophisticated instructional practices
• Concrete exemplars of high-quality instructional practice

District Plan: Teacher Networks
• 1-2 mathematics teachers in each school receive additional intensive mathematics professional development
• Lead mathematics teachers
• Facilitate biweekly or monthly teacher study group meetings

Findings and Recommendations :
Teacher Networks
• Quality of professional development for lead teachers high
• Does not focus specifically on teaching underserved groups -- English language learners
(ELLs)
• Additional professional development for lead teachers on:
• Teaching language in the context of mathematics
-- ELLs

Findings and Recommendations :
Teacher Networks
• Collaboration between isolated pairs of mathematics teachers in some schools
• Typically low depth
• No opportunities for lead teachers to share what they are learning in most schools
• Common planning time for mathematics teachers
• Additional professional development for lead teachers on:
• Process of supporting colleagues’ learning
• Organizing the content of a study group’s work

Findings and Recommendations :
Teacher Networks
• At least one mathematics teacher in each school with a sophisticated view of highquality mathematics instruction
• Principals selected teachers for additional professional development
• District policy: criteria for selecting lead mathematics teachers

Conjecture: Shared Vision of High
Quality Mathematics Instruction
• Instructional goals -- what students should know and be able to do mathematically
• How students' development of these forms of mathematical knowing can be supported

Conjecture: Shared Vision of High
Quality Mathematics Instruction
• Coordination between district administrative units
• Curriculum and Instruction
• Leadership
• Research and Evaluation
• English Language Learners
• Special Education

Conjecture: Shared Vision of High
Quality Mathematics Instruction
• Occupational groups: Mathematics teachers, principals, district mathematics specialists, district leadership specialists, …
• Differences in:
• Responsibilities
• Practices
• Professional affiliations (and professional identities)

Conjecture: Brokers
• Participate at least peripherally in the activities of two or more groups
• Can bridge between differing agendas for mathematics instruction

District Plan: Shared Instructional
Vision
• Curriculum Cabinet -- heads of all district units + area superintendents
• Professional development in instructional leadership for all principals
• Not content specific
• Intellectually-demanding tasks
• Maintain the challenge of the tasks as they are enacted in the classroom
• Compatible with district goals for mathematics instruction

Findings and Recommendations : Shared
Instructional Vision
• District leaders: Inconsistent visions + not specific to mathematics
• Form rather than function views
• Area superintendents participate in mathematics professional development with lead teachers
• Expertise in Curriculum Cabinet
• Support alignment between Curriculum and Instruction, and Leadership
• Brokers between district leaders and principals

Findings and Recommendations : Shared
Instructional Vision
• Principals: Not specific to mathematics
• Form rather than function views
• Teachers: At least one mathematics teacher in each school with a sophisticated view of high-quality mathematics instruction
• Few formal opportunities for principals to draw on or learn from expert teachers

Findings and Recommendations : Shared
Instructional Vision
• Principals share leadership of mathematics study groups with lead teachers
• Principals gain access to mathematics expertise in their schools
• Brokers between mathematics teachers and school/district leaders
• Legitimize work of lead teachers
• Lead teachers can focus on content-specific aspects of study group activities

Conjecture: Mutual Accountability
• School leaders hold mathematics teachers accountable for developing high-quality instructional practices
• School leaders are accountable to mathematics teachers (and district leaders) for supporting teachers’ learning

Conjecture: Leadership Content
Knowledge (in Mathematics)
• Enables school and district leaders to:
• Recognize high-quality mathematics instruction
• Support teachers’ learning directly
• Organize the conditions for ongoing learning of school and district staff
(Stein & Nelson)

Conjecture: Leadership Content
Knowledge
• Principals require a relatively deep understanding of:
• Mathematical knowledge for teaching
• How students learn mathematics
• What is known about how to teach mathematics effectively
• Teachers-as-learners and effective ways of teaching teachers

Conjecture: Leadership Content
Knowledge
• Distributed across formal and informal leaders
• Lead mathematics teachers
• Accomplished teachers as informal instructional leaders
• Principal instructional leadership expertise involves recognizing and capitalizing on mathematics teachers’ expertise

District Plan: Mutual Accountability
• Professional development in instructional leadership for all principals
• In classrooms observing instruction for two hours each day
• Use developing understanding of (content-free) high-quality instruction to:
• Assess quality of instruction and give feedback to teachers
• Organize school-level teacher professional development
• Develop school improvement plans

Findings and Recommendations : Mutual
Accountability
• Most principals do not view themselves as instructional leaders
• Most principals are spending only limited time in classrooms
• Inconsistent messages from district leaders -- not aware that district leaders expect them to be in classrooms
• District leaders need to communicate expectations for what it means to be an instructional leader clearly and consistently
• Hold principals accountable for supporting mathematics teachers in improving their instructional practices

Findings and Recommendations : Mutual
Accountability
• Most Principals have developed form rather than function views of high-quality mathematics instruction
• Feedback to teachers focuses on surface level features of instruction (e.g., arranging students in groups)
• Most principals are not organizing school-based professional development for mathematics teachers
• No supports for principals as instructional leaders beyond professional development

Findings and Recommendations : Mutual
Accountability
• Principals participate in at least a portion of mathematics professional development with lead teachers
• Principals share the leadership of mathematics study groups
• Area superintendents provide guidance on:
• Providing constructive feedback to teachers
• Organizing school-based professional development

Findings and Recommendations : Mutual
Accountability
• Generic classroom observation form specifies “promotion of innovative teaching methods”
• Redesign observation form to reflect district vision of high-quality mathematics instruction

Summary: Conjectured Support
Structures
• Teacher networks
• Time for collaboration
• Access to expertise
• Shared instructional vision
• Brokers
• Mutual accountability
• Leadership content knowledge

Current and Next Steps
• Fall 2009:
• Document whether districts actually act on the basis of our feedback
• January-March 2009:
• Document the consequences of any adjustments
• May 2009:
• Second round of feedback to districts

Research Agenda
• Test, revise, and modify conjectures about relationships between:
• Changes in school and district support structures
• Improvement in mathematics teachers’ instructional practices
• Student achievement

Research Agenda
• Refine conjectures:
• Identifying additional support structures
• Clarifying relationships between support structures
• Specifying the conditions under which particular support structures are important

Teachers’ Access to Expertise:
Local and External Views of Expertise
• Local views: Who teachers identify as experts
• Criteria for what counts as instructional expertise
• External views: Teachers we identified as experts:
• District views of high-quality mathematics instruction
• Research literature on mathematics learning and teaching

Teachers’ Access to Expertise:
The Role of the Principal
• Teachers’ access to expertise
• Teacher networks, mathematics coaches, district math specialists, external expertise
• Principals’ practices
• The how of instructional leadership
• Principals’ knowledge-of-practice
• Vision of high-quality mathematics instruction
• (Suppositions about process of teacher learning)

Policy and Learning
• Policy
• Local, state, and national policies intentionally designed to influence teachers’ classroom practices
• Mathematics education
• Professional development and instructional materials intentionally designed to influence teachers’ classroom practices

Policy Research
• The outcomes of specific policies
• The process by which particular policies are implemented
• No position on what high-quality instruction looks like

Mathematics Education
• Students’ and teachers’ learning
• Classroom in an institutional vacuum

Learning Policy
• Formulate and refine policies by building on research on learning and teaching
• Frame instructional improvement as a problem of organizational learning for schools and districts