1. science
2. mathematics

DoDEA Mathematics Standards Gap Analysis
Alignment of DoDEA’s College and Career Ready Standards for Mathematics with
DoDEA Academic Standards
AIR: AMERICAN INSTITUTES FOR RESEARCH
Tad Johnston, Robin Bzura, Steve Leinwand, Beth Ratway, Tori Cirks, Asta Svedkauskaite
The Department of Defense Education Activity
Sarah Koebley, Jeffery Epting, Gelonda Martin, Matt Syarto, Joseph Skorupski, Elise Rosch, Judy McGimsey, Wendy Sancho, Patti McNaughton, Challice Rickard
DoDEA CCRSM Gap Analysis
Introduction
The Department of Defense Education Activity (DoDEA) is in the process of transitioning from their current academic standards to the College and Career
Ready Standards for Mathematics (CCRSM). 1 As part of its mission to “Educate, Engage, and Empower each student to succeed in a dynamic world,”
DoDEA contracted with American Institutes for Research (AIR) to conduct a gap analysis of its current standards and curriculum to inform the
implementation process of the College and Career Ready Standards. This document contains key highlights from the original AIR document including the
alignment of the Standards of Mathematical Practice to the DoDEA Mathematics Process Standards. The specific Content Standards alignment may be
found in Appendix A of this document.
Overall Findings
This section provides an overview of key findings. General trends across grades and the match between the DoDEA Mathematics Process Standards and
CCRSM Mathematical Practices are described.
Overall Grades K–8 findings are as follows:
•
Overall Number. The overall number of DoDEA standards (N = 315) and CCRSM standards (N = 317) are similar.
•
Grade-Level Alignment. Alignment is stronger at Grades K–3 (11 percent of CCRSM standards do not have a DoDEA match) than at Grades 4–8
(26 percent of CCRSM standards do not have a DoDEA match).
Overall Grades 9–12 findings are as follows:
•
Overall CCRSM Match. 30 percent of the CCRSM standards are not matched to a DoDEA standard or standards.
•
High or Partial CCRSM Match. Of the 132 standards that are either highly matched or partially matched by DoDEA standards, 63 (47 percent)
require a course beyond the Algebra 1, Geometry, Algebra 2 sequence for the level of alignment reported.
Across the grades, there are many partial alignments because the specific mathematical expectations for conceptual understanding, specified levels of
procedural skill and fluency, methods of representation (e.g., double number lines or ratio tables), and reference to applications (real-world or
mathematical problems) in the CCRSM standards were not found in the DoDEA standards. This same degree of specificity in the CCRSM standards also
resulted in non-matches with DoDEA standards if the specific understanding or skill level was listed in a separate CCRSM standard and was not found in
the set of DoDEA standards examined.
1
The abbreviation “CCRSM” refers to DoDEA’s College and Career Ready Standards for Mathematics, which are derived from the Common Core State Standards. The
specific standards can be found at http://www.corestandards.org/read-the-standards/
Alignment of DoDEA CCRSM with the DoDEA Academic Standards
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DoDEA CCRSM Gap Analysis
Although the guiding question was how the set of DoDEA standards provided coverage of the CCRSM standards, many DoDEA standards did not align
with any CCRSM standards. As DoDEA transitions to the CCRSM standards, these DoDEA standards may not be needed. A list of K-5 DoDEA standards not
used in the alignment and a table showing the grade level at which DoDEA K–5 standards were aligned to CCRSM standards is included in the Gap
Analysis spreadsheet.
There are some notable trends in this analysis. At the K–5 level, probability does not appear in the CCRSM standards until Grade 7; DoDEA includes
probability in Grades 2–6, which is why many DoDEA probability standards are not aligned with the CCRSM standards. Similarly, in early grades, patterns
and ordinal numbers, found in DoDEA, are not found in the CCRSM. Many, and sometimes all, of the DoDEA geometry standards in Grades 2, 4, and 7
also are not aligned.
Findings: CCRSM Standards for Mathematical Practice and DoDEA Process Standards
Both the DoDEA standards and the CCSSM begin with learning expectations for students across all grade levels and all for all content areas of
mathematics. For DoDEA Mathematics, these are the Process Standards and for the CCRSM State Standards for Mathematics, these are the
Mathematical Practices. The DoDEA Process Standards describe goals for the processes of problem solving, reasoning and proof, connections,
communication, and representation. For the CCRSM, the Standards for Mathematical Practice “describe varieties of expertise that mathematics
educators at all levels should seek to develop in their students” (National Governors Association Center for Best Practices & Council of Chief State School
Officers, 2010, p. 6). The full statement of the mathematical practices was considered and compared with the bullets in the DoDEA Process Standards to
determine the level of support for the CCRSM Mathematical Practices provided by the DoDEA Mathematics Process Standards (DoDEA Mathematics
Standards Pre-Kindergarten Through Grade 8, 2009, p. 4)
Mathematical Practice 1
“Make sense of problems and persevere in solving them,” has strong association with the DoDEA Process Standards for Problem Solving
(solve problems that arise in mathematics and in other contexts, apply and adapt a variety of appropriate strategies to solve problems,
monitor and reflect on the process of mathematical problem solving), Reasoning and Proof (make and investigate mathematical
conjectures), and Representation (select, apply, and translate among mathematical representations to solve problems; use
representations to model and interpret physical, social, and mathematical phenomena) (National Governors Association Center for Best
Practices & Council of Chief State School Officers, 2010, p. 6). The quality of perseverance is not mentioned in the DoDEA standards.
Mathematical Practice 2
“Reason abstractly and quantitatively,” is supported partially by the DoDEA Process Standards (National Governors Association Center
for Best Practices & Council of Chief State School Officers, 2010, p. 6). This practice is associated with the DoDEA Process Standards for
Problem Solving (apply and adapt a variety of appropriate strategies to solve problems), Reasoning and Proof (select and use various
types of reasoning and methods of proof), and Representation (select, apply, and translate among mathematical representations to
solve problems). The habits of quantitative reasoning, however, were not found in the Process Standards. Other key components of
Mathematical Practice 2 not included in the DoDEA standards are the ability to decontextualize (abstracting a situation to treat it as
mathematical objects) and to contextualize (check that the mathematical work still accurately represents the real situation).
Alignment of DoDEA CCRSM with the DoDEA Academic Standards
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DoDEA CCRSM Gap Analysis
Mathematical Practice 3
“Construct viable arguments and critique the reasoning of others,” is supported well by the DoDEA Process Standards (National
Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, pp. 6–7). Strong association was seen
with Reasoning and Proof (make and investigate mathematical conjectures, develop and evaluate mathematical arguments and proofs,
select and use various types of reasoning and methods of proof), Communication (analyze and evaluate the mathematical thinking and
strategies of others), and Representation (use representations to model and interpret physical, social, and mathematical phenomena).
It should be noted that counterexample as a method of argument and proof is not specifically mentioned in the DoDEA standards.
Mathematical Practice 4
“Model with mathematics,” is also supported well by the DoDEA Process Standards (National Governors Association Center for Best
Practices & Council of Chief State School Officers, 2010, p. 7). Strong association was seen with Problem Solving (solve problems that
arise in mathematics and in other contexts, monitor and reflect on the process of mathematical problem solving), Connections
(recognize and use connections among mathematical ideas, recognize and apply mathematics in contexts outside of mathematics), and
Representation (select, apply, and translate among mathematical representations to solve problems; use representations to model and
interpret physical, social, and mathematical phenomena).
Mathematical Practice 5
“Use appropriate tools strategically,” is not supported by the DoDEA Process Standards, which do not address the use of tools,
technology, or identification of mathematical resources (National Governors Association Center for Best Practices & Council of Chief
State School Officers, 2010, p. 7).
Mathematical Practice 6
“Attend to precision,” is supported partially by the DoDEA Process Standards (National Governors Association Center for Best Practices
& Council of Chief State School Officers, 2010, p. 7). Association was seen with the Process Standard for Communication (communicate
their mathematical thinking coherently and clearly to peers, teachers, and others; use the language of mathematics to express
mathematical ideas precisely), but the emphasis on units, accurate and efficient calculation, and expressing numbers to appropriate
levels of precision was not seen in the DoDEA Process Standards.
Mathematical Practice 7
“Look for and make use of structure,” also is supported partially by the DoDEA Process Standards (National Governors Association
Center for Best Practices & Council of Chief State School Officers, 2010, p. 8). While the DoDEA Process Standards support the parts of
the CCSSM practice related to seeing how previous knowledge may apply to a current problem, choosing topic-specific strategies, and
stepping “back for an overview and shift perspective,” the DoDEA Process Standards do not address discerning patterns and structure
or composing and decomposing mathematical objects specifically. Association was seen with Problem Solving (apply and adapt a
variety of appropriate strategies to solve problems, monitor and reflect on the process of mathematical problem solving) and
Connections (recognize and use connections among mathematical ideas).
Mathematical Practice 8
“Look for and express regularity in repeated reasoning,” is supported, but not fully, by the DoDEA Process Standards (National
Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, p. 8). Association was seen with Problem
Solving (build new mathematical knowledge through problem solving, solve problems that arise in mathematics and in other contexts,
monitor and reflect on the process of mathematical problem solving), Reasoning and Proof (select and use various types of reasoning
and methods of proof), and Connections (recognize and use connections among mathematical ideas). Generalization and abstraction
from experiences were considered critical to the practice and are not seen specifically in the DoDEA Process Standards.
Alignment of DoDEA CCRSM with the DoDEA Academic Standards
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DoDEA CCRSM Gap Analysis
Findings – Content Standards
Although there were many partial alignments between DODEA standards and the CCRSM standards, the CCRSM Standards include increased emphasis
on conceptual understanding, specific levels of procedural skill and fluency, representational models (e.g. double number lines, ratio tables) and realworld applications. The Gap Analysis findings are summarized below by grade level and course with an emphasis on the standards which are either
partially aligned (at the existing grade level and other grade levels) or not aligned at all. These are the standards on which teacher professional
development will need to focus the most.
Grade Band
Standards Partially
or Not Aligned
K-2
56/77 (73%)
3-5
91/144 (80%)
6-8
116/126 (92%)
9 - 12
157/192 (82%)
K - 12 Standards for
Mathematical
Practice
Notes/Recommendations
7of the 10 standards in Kindergarten's Counting/Cardinality domain are not fully aligned. 88% of the
Operations/Algebraic Thinking domains are not supported in Grades 1 and 2. With these standards as a crucial
foundation for number sense throughout students' mathematical careers, implementation should start with these
domains (including Number/Operations in Base Ten) focused on counting, addition and subtraction.
There are many non-aligned standards in the domain of Fractions (89%) with the overall trend being that
concepts appear at earlier grade levels (with the introduction to unit fractions occurring at Grade 3). As this
number concept is crucial to later success in mathematics, focused early implementation of this domain as well as
on the more rigorous standards of multiplication/division is recommended.
Grades 6 - 8 have the greatest number of standards (K-12) which are non-aligned or partially aligned. Ratio and
Proportional Reasoning, a foundational concept for algebra and geometry, has only 1 standard (out of 14 - or 7%)
which is fully aligned for Grades 6 and 7. CCRSM's emphasis on algebraic thinking in middle school is evident in
that the domain of Expressions and Equations has the most total standards in MS (31); DODEA is fully aligned to
only 6 of those standards (19%).
The high school CCRSM standards have the greatest number (30%) which are not matched to any DODEA
standard. Of the 132 that are matched, 63 standards (47%) require a course beyond the
Algebra1/Geometry/Algebra2 sequence for the level of alignment reported. Evidence of large gaps in standards
were found in the following areas: seeing structure in expressions, reasoning with and creating equations,
interpreting functions, building functions, congruence, similarity, circles, geometric modeling, all domains of
statistics and probability. The overall emphasis in CCRSM on modeling and the use of technology at the high
school level is not found in DODEA's standards.
Missing in DODEA standards: perseverance in problem solving (MP1); quantitative reasoning and contextualizing/decontextualizing
problem situations (MP2); using counterexamples and critiquing the math reasoning of others (MP3); creating mathematical models
(MP4); using appropriate tools strategically (MP5); attend to precision (MP6); look for and make use of structure (MP7); and generalizing
and abstracting mathematical properties (MP8).
Alignment of DoDEA CCRSM with the DoDEA Academic Standards
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DoDEA CCRSM Gap Analysis
Innovation Configuration (IC) Maps for Measuring Implementation Progress
The AIR team developed an innovation configuration map as a tool to help schools measure progress towards systemic
CCRSM implementation. Innovation configuration (IC) maps, introduced by Hord, Hirsh, and Roy (2005), can create
common language around standards implementation. They identify research-based key attributes of the implementation,
the highest levels of desired outcomes, and the incremental variables that lead to those levels.
At the classroom level, the IC map, adapted from Hord et al. (2005), provides a practical rubric for practitioners to
analyze the current status and assess implementation progress (innovating, implementing, progressing, and beginning).
These IC maps, as tools, can be used by different individuals and for different purposes. Instructional teams can use the
maps to determine variations within and across classrooms. Individual classroom teachers can use these maps to reflect
on their own instruction and then bring the maps to the grade- level
instructional teams to discuss. When used this way, the tool should
not act as a rating tool. Instead, it acts as a tool to facilitate reflection
and teachers using it should be as candid as possible when
determining level of performance. Every classroom and grade level
will be at different stages in different components.
It is expected that many teams will be in the early stages of implementation as DoDEA is just
beginning to implement the CCRSM standards. The tools can be used by teams of teachers (that may
or may not have a designated facilitator) and/or individual teachers to facilitate reflection and
identify gaps in implementation. They could also be used at the district level, potentially in combination with other data, to assess implementation and
determine next steps.
Alignment of DoDEA CCRSM with the DoDEA Academic Standards
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DoDEA CCRSM Gap Analysis
Classroom-Level IC Map: Curriculum
Successful implementation of the CCRSM Standards that increases levels of student learning requires a curriculum to be a well-defined set of research-based
experiences in which students and instructors are engaged to attain understanding and achievement of outcomes and expectations aligned to the standards.
4 (Innovating)
3 (Implementing)
2 (Progressing)
1 (Beginning)
Identifies and reviews resources that
identify gaps and translate the standards
to determine the underlying knowledge,
skills, fluency, conceptual understanding
and procedural knowledge in the
standards. Identifies tools and specific
examples that articulate the
characteristics of successful academic
performance and that can be used to align
and improve curricula.
Reviews the standards to begin to
determine underlying knowledge,
skills, fluency, conceptual
understanding and procedural
knowledge in the standards.
Reviews examples that articulate
the characteristics of successful
academic performance and that can
be used to align and improve
curricula.
Desired Outcome: Translate the standards to support implementation.
Collaborates with colleagues to identify
gaps and translate the standards to help
ensure a clear and common understanding
and to determine the underlying
knowledge, skills, fluency, conceptual
understanding and procedural knowledge
in the standards. Uses tools and specific
examples that clearly articulate the
characteristics of successful academic
performance and that can be used to align
and improve curricula.
Begins to identify gaps and translate the
standards to help ensure a clear and common
understanding and to determine the
underlying knowledge, skills, fluency,
conceptual understanding and procedural
knowledge in the standards. Identifies and
shares tools and specific examples that
clearly articulate the characteristics of
successful academic performance and that
can be used to align and improve curricula.
Desired Outcome: Use tools and evaluations to help identify and select appropriate resources and materials.
Works collaboratively to clearly define the
intended learner results and effective
learning strategies required of the
standards, and uses a variety of existing
tools to help identify and select appropriate
materials based on rigorous research-based
practices.
Uses a variety of existing tools to help
identify and select appropriate materials
based on rigorous research-based practices
that link to student learning, clearly define
the intended learner results, and
demonstrate connections to effective
learning strategies required of the standards.
Identifies a variety of appropriate
materials that say they link to student
learning, clearly define the intended
learner results, and demonstrate
connections to effective learning
strategies required of the standards.
Reviews existing materials to begin
to identify if the materials are
connected to the intended learner
results and learning strategies
required of the standards.
Desired Outcome: Create elaborations of the standards to help students envision how they play out in classrooms.
Engages with colleagues to develop
rigorous instructional units and curricular
tools that identify real-world applications of
the standards, including cross- curricular
connections, cognitive expectations, and
sequencing to create opportunities for
moving all students to higher levels of
academic growth as required by the
standards.
Engages with colleagues to develop
instructional units and curricular tools that
include cross-curricular connections,
cognitive expectations, and sequencing to
create opportunities for moving students to
higher levels of academic growth as required
by the standards.
Alignment of DoDEA CCRSM with the DoDEA Academic Standards
Develops instructional units and
curricular tools that offer opportunities
for students to engage in the cognitive
expectations required by the standards.
Reviews externally created
instructional units and curricular
tools that offer opportunities for
students to engage in the cognitive
expectations required by the
standards.
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DoDEA CCRSM Gap Analysis
Classroom-Level IC Map: Instruction
Successful implementation of the CCRSM Standards that increases levels of student learning requires instruction based on a clear vision for teaching and
learning, portraying a clear commitment to learning for understanding.
4 (Innovating)
3 (Implementing)
2 (Progressing)
1 (Beginning)
Outlines a set of explicit principles for
learning and instructional design based
on research that ensures meaningful
learning opportunities for students.
Outlines a set of instructional strategies
based on previous practices and
experiences that connect directly to the
standards.
Desired Outcome: Create a clear vision for teaching and learning.
Creates a vision for teaching and
learning that shows a commitment to
the expectation that all students can
achieve success, and outlines a set of
explicit principles for learning and
instructional design based on research
that ensures meaningful learning
opportunities for each student.
Creates a vision for teaching and
learning so that all students can achieve
success, and outlines a set of principles
for learning and instructional design
based on research that ensures
meaningful learning opportunities for
each student.
Desired Outcome: Use research-based instructional practices that are aligned with the vision of the standards.
Uses studies and reports of researchbased instructional practices that are
linked to student achievement, and
provides clear definitions and examples
of instructional practices to create
opportunities for moving all students to
higher levels of academic growth as
required by the standards.
Uses studies and reports of researchbased instructional practices that are
linked to student achievement to
create opportunities for moving all
students to higher levels of academic
growth as required by the standards.
Uses reports of research-based
instructional practices to create
opportunities for moving students to
higher levels of academic growth as
required by the standards.
Uses existing examples of instructional
practices to create opportunities for
students to reach the academic
standards.
Desired Outcome: Develop instructional models focusing on high-quality, standards-based instruction.
Collaboratively develops and integrates
instructional strategies using a wide
array of research focusing on highquality, standards-based instruction.
Develops and integrates instructional
strategies using a wide array of
research connected to standards-based
instruction.
Alignment of DoDEA CCRSM with the DoDEA Academic Standards
Develops and integrates instructional
strategies focusing on high-quality,
standards-based instruction.
Develops instructional strategies using
a wide array of resources connected to
standards-based instruction.
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DoDEA CCRSM Gap Analysis
Classroom-Level IC Map: Assessment
Successful implementation of the CCRSM State Standards that increases levels of student learning requires assessments that include the processes
used to measure student mastery of the standards and the delivery of curriculum experiences through both formative and summative methods
(Pellegrino, 2006).
4 (Innovating)
3 (Implementing)
2 (Progressing)
1 (Beginning)
Desired Outcome: Engage in training programs to build summative, benchmark, interim, and formative assessments aligned to the standards.
Assesses the impact of the standards on student
learning by developing effective assessment
measures of student learning and uses the results
to inform and improve teaching and learning.
Engages with colleagues to study innovative
assessment practices, multiple assessment
strategies, recognizing positive results and
refining programs as needed.
Assesses the impact of the
standards on student learning by
developing assessments to
measure student learning and uses
the results to improve and inform
teaching and learning.
Alignment of DoDEA CCRSM with the DoDEA Academic Standards
Assesses the impact of the
standards on student learning by
developing assessments measures
of student learning.
Assesses the impact of the standards
on student learning.
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