The Vermont Mathematics Partnership Ongoing
Assessment Project (OGAP): A Cognitively
Based Instructional Intervention in Mathematics
Bob Laird, Vermont Mathematics Institute ([email protected] )
Marge Petit, Marge Petit Consulting, MPC ([email protected] )
CPRE CCII Meeting
February 2008
Vermont Mathematics Partnership
www.vermontmathematics.org
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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In this Introduction:
•Present a broad overview of the Ongoing
Assessment Project (OGAP)
•Introduce you to important aspects of the
project
•Share some of what we are learning
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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OGAP:
Responding to Two Needs
1) To improve student learning in mathematics for all
students as it relates to Vermont Standards and
Grade Level Expectations (GLEs)
2) For teachers to obtain quality instructional
information as students are developing their
understanding of concepts so that interventions for
a class as a whole or for individuals can be made
“on time.”
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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VMP OGAP Design Team
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Leslie Ercole, VMP
Linda Gilbert, Dotham Brook School
Kendra Gorton, Milton Elementary School
Steph Hockenbury, Chamberlin School
Beth Hulbert, Barre City Elementary and
Middle School
Amy Johnson, Milton Elementary School
Bob Laird, VMP
Ted Marsden, Norwich University
Karen Moylan, Former VMP
Cathy Newton, Dotham Brook School
Susan Ojala, Vermont Mathematics Initiative
Nancy Pollack, Chittenden East
Marge Petit, Marge Petit Consulting, MPC
Regina Quinn, VMP
Loree Silvis, VMP
Krisan Stone, VMP
Corrie Sweet, Former VMP
Tracy Thompson, Ottauquechee School
Jean Ward, Bennington Rutland Supervisory
Union
Rebecca Young, Hardwick Schools
Active OGAP National Advisory Board
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Mary Lindquist, Callaway Professor of
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Mathematics Education, Emeritus; Past
President of the National Council of Teachers of
Mathematics
Ed Silver, University of Michigan
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Judith Zawojewski, Illinois Institute of
Technology
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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OGAP is an intentional and systematic approach to
formative assessment in mathematics involving:
• Gathering information about preexisting knowledge through the use
of a pre-assessment;
• Analysis of pre-assessment to guide
unit planning; and
• A continuous and intentional
system of instructing, probing with
instructionally embedded questions,
analysis, and instructional
modification.
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
Grades 2 - 8
•Fractions
•Multiplicative
reasoning
•Proportionality
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In Place and In Use
for all three mathematical topics
• Item banks and pre-assessments
• Tools and strategies to analyze student
work
• Professional development workshop
materials and resources to communicate
research and support the use of OGAP
formative assessment system
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Development and Implementation
Model Based on…
• Distillation of hundreds of research
articles and synthesis of research into
frameworks
• Interaction with over 300 educators
(over 6000 students) in Vermont and
Alabama with OGAP materials and
resources (and numbers are growing)
• Analysis of teacher logs linked to
student work archives
• Analysis of over 40,000 pieces of
student work
• Analysis of teacher action research
projects
• Surveys, interviews, feedback forms
• Advice from National Advisory Board
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
Maps not to scale
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OGAP Theory of Action
Educators make more effective instructional decisions that
result in improved student learning when they…
• Are knowledgeable about the cognitive research on how
students develop understanding of specific mathematics
concepts and where students commonly encounter difficulties;
• Access tools and strategies sensitive to the cognitive
research that allows them to intentionally and systematically
monitor student learning prior to and during instruction;
and
• Receive professional development that supports their
classroom application of this formative assessment system.
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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It is not formative assessment alone OR
knowledge of cognitive research
alone…
…but the marriage of the
two that empowers teachers
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Hundreds of research
articles distilled into a
frameworks and used
In design of materials
• formative assessment items
(hundreds)
• professional development
materials (case studies,
activities, essays)
In work with educators
• analyze student work
• inform instructional decisions
• help understand the purposes of
activities in mathematics
programs
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Teachers say – that knowledge of cognitive research coupled
with tools and resources sensitive to the research helps them …
• Understand the purposes of activities in
math programs;
• Understand evidence in student work used
to inform instruction;
• Strengthen and focus first wave instruction;
• Respond to evidence in student work as
instruction proceeds.
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Draft
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Going beyond
celebrating different
strategies TO…
…understanding the
instructional implications
of the strategies and
taking action
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Example of Non-fractional Reasoning
There are some candies in a dish.
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5 of the candies are chocolate.
3
of the candies are peppermint.
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Are there more chocolate candies or peppermint candies in
the dish?
Use words, pictures, or diagrams to explain your answer.
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Example of a Fractional Strategy with Error
Five-twelfths of the gym was used for a kickball game.
One-third of the same gym was used for a football game.
Which game used more of the gym?
Use words, pictures, or diagrams to explain your answer.
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Example of a Transitional Strategy
There are some candies in a dish.
2
5 of the candies are chocolate.
3
of the candies are peppermint.
10
Are there more chocolate candies or peppermint candies in
the dish?
Use words, pictures, or diagrams to explain your answer.
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Example of Flexible a Fractional Strategy
There are some candies in a dish.
2
5 of the candies are chocolate.
3
of the candies are peppermint.
10
Are there more chocolate candies or peppermint candies in
the dish?
Use words, pictures, or diagrams to explain your answer.
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Examples of teacher interventions (response to
inappropriate whole number reasoning and lack of use of models)
• Use modeling to build concepts
• Emphasis on number line
• Emphasis on relative magnitude of
fractions using modeling and other
reasoning strategies
VMP OGAP Exploratory Studies (2004,
2005) and 2006-2007 Roll-out
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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OGAP and Coherence to Curriculum and
Instruction
• Places focus on the development of big mathematical
ideas across mathematical topics;
• Provides knowledge of how these ideas develop and
interrelate, and common errors or misconceptions that
might interfere with learning new concepts or solving
problems;
• Provides tools, strategies, and resources based on
cognitive research to probe for understanding prior to
and during instruction.
• Provides actionable information when it is needed.
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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VMP is a targeted
Math & Science
Partnership funded by
the National Science
Foundation
&
the U.S. Department of
Education
National Science Foundation,
grant award number EHR – 0227057
and U.S. Department of Education,
grant award number S366A020002
Mathematicians and Educators
working together to help all
Vermont children
succeed in mathematics
www.vermontmathematics.org
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Example of Non-fractional Reasoning
Inappropriate
whole number
reasoning
According to research, some students
may see a fraction as two whole
numbers (e.g., ¾ as a 3 and 4)
inappropriately using whole number
reasoning, not reasoning with a
fraction as a single quantity.
(Behr, M., Post, T.,
Lesh, R., and Silver, E. (1983); Behr, Wachsmuth and Post, (1984); VMP OGAP Study (2005))
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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2007 Action Research Data
(Redmen, C. (2007))
Mean Number of Responses out of 23
Grade 4 OGAP Pre to Post: Incidence of Inappropriate Whole Number
Reasoning
12
10.06
10
7.1
8
Pre
6
4
4.48
Post
2.38
2
0
Treatment(n=32)
Control (n=10)
Cohen’s d Effect Size = 0.77
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Non-fractional Reasoning
Fractional Strategy
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Non-fractional Reasoning
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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Non-fractional Reasoning
THe Vermont Mathematics Partnership funded by the USDOE(S366A020002) and NSF ((EHR-0227057)
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