Mathematics Assessment
Framework
Grades 3–8
For Assessments Beginning School Year 2012–2013
Prepared by the
New Mexico Public Education Department
Assessment and Accountability Division
300 Don Gaspar
Santa Fe, New Mexico
April 2012
Susana Martinez, Governor of the State of New Mexico
Hanna Skandera, Secretary-Designate of the New Mexico
Public Education Department
Acknowledgements
Dr. Pete Goldschmidt is Director of Assessment and Accountability at the New Mexico
Public Education Department (PED). Dr. Tom Dauphinee is the division’s Deputy Director.
Dr. Dauphinee led the development of the Assessment Frameworks in conjunction with the
Assessment Frameworks Development Workgroup and with the assistance at the PED of
Claudia Ahlstrom, State Math Specialist; Diana Jaramillo, Education Administrator for the
Assessment Bureau; and Melinda Webster, Reading Program Director.
Members of the Assessment Frameworks Development Workgroup included the following:
 Karon Axtell, Carlsbad Municipal Schools
 Lorene Beckstead, Los Alamos Public Schools
 Elizabeth Jacome, Rio Rancho Public Schools
 JoRaye Jenkins, Carlsbad Municipal Schools
 Linda Kinane, Albuquerque Public Schools
 Amanda Knott, Carlsbad Municipal Schools
 Dora (Mae) Montaño, Rio Rancho Public Schools
 Linda Pehr, Alamogordo Public Schools
Dr. Howard T. Everson, professor of education and psychology at the City University of New
York, served as senior technical and policy advisor. New Mexico’s statewide educational
foundation, the Advanced Programs Initiative (API), facilitated the project and wrote the
report with the support of Marybeth Schubert, Eilani Gerstner, and Melissa Wauneka.
The University of New Mexico College of Education and Dean Richard Howell provided
invaluable logistical help by hosting the Workgroup at the college on February 27–29, 2012.
S u m m a r y
E x e c u t i v e
New Mexico and 45 other states have adopted Common Core State
Standards (CCSS) for public schools, establishing new guidelines for student learning
Executive
Summary
that are internationally competitive. Developed over many years, tested and proven to
be effective, these new learning standards draw on research on how students learn
and how best to prepare them for college and the increasingly competitive job market.
To spur its implementation of CCSS, New Mexico is a member of the Partnership for
Assessment of Readiness for College and Careers (PARCC)—a consortium of 24
states working together to develop a common assessment system. By 2015, all
statewide accountability tests in New Mexico will be developed and administered by
PARCC. In other words, New Mexico students will be taking the same standards
based assessments as students in 23 other states. These assessments will be
delivered and taken via computer.
The New Mexico Public Education Department (PED) is responsible for developing
assessment frameworks for statewide testing during the transition to PARCC. The
purpose of the Mathematics Assessment Framework is to inform educators, test
developers, the public, and policy makers of the topics in the CCSS for math that will
be emphasized over the next two years in New Mexico's Standards Based
Assessment (SBA).
New Mexico is seeking to better prepare teachers and students to meet the
heightened expectations of the new learning standards and the PARCC assessments
by steadily raising the rigor of the state’s standards based assessment. In school year
2012–2013, the SBA will test students on the New Mexico state standards for grades
4–8, 10 and 11, and there will be a Bridge Assessment for grade 3 that is dually
aligned to CCSS and the New Mexico state standards. In 2013–2014, there will be a
“bridge” SBA for students in all tested grades (3–8, 10, and 11) that will look and feel
more like the PARCC assessment.
The Bridge Assessments will follow the state’s current proficiency ratings—Beginning
Step, Nearing Proficiency, Proficient, Advanced—for reporting scores based on New
Mexico Academic Content Standards as outlined in the New Mexico Standards Based
Assessment: Standard Setting Report (2011). School Grades in the A-F
accountability system will continue to be based on growth in student scores using
New Mexico Academic Content Standards. The Bridge Assessment will include
newly developed items aligned with CCSS starting in 2014, but those items will not
count towards school accountability grades. The PED will issue a separate report to
districts about student proficiency on CCSS items and tasks.
The CCSS value depth of knowledge over breadth of knowledge. This means that
New Mexico teachers should expect to teach less content but with more complexity.
When designing assignments, teachers should keep in mind the over-arching goals of
the CCSS for math, which are to constantly integrate the mathematical practices
throughout students’ learning and help students become life-long critical thinkers and
problem solvers. These goals should be attained through an increased focus on
algebraic or “abstract” thinking, fluency in number and operations, and greater
understanding of proportional relationships and probability.
Table of Contents
I.
Introduction
1
II.
About the Common Core State Standards
Organization
Mathematical Practices that Affect All Grades
Learning Progressions
Six “Shifts” in Mathematics
College and Career Readiness
2
2
3
3
4
5
III.
Design Principles for the Bridge Assessments
What is the Bridge Assessment Framework? Why is it Needed?
Design Goals
Impact on Instruction
Use of Technology
5
5
6
6
7
IV.
Reporting and Accountability
Policy and Definitions
Technology
Scaffolding
Complexity
Item Formats
Achievement Levels
SBA Results and the A-F Accountability System
8
8
8
8
9
9
9
10
V.
Critical Skills
Grades 3–5
Operations and Algebraic Thinking
Numbers and Operations in Base Ten
Numbers and Operations-Fractions
Measurement and Data
Geometry
10
Grades 6–8
Ratios and Proportional Relationships
The Number System
Expressions and Equations
Geometry
Statistics and Probability
Functions (Grade 8)
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
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11
12
12
12
13
13
13
14
14
14
i
Table of Contents
VI.
Conclusion
15
VII.
Appendices
Reference Tables
Bibliography
16
ii
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
I. Introduction
New Mexico and 45 other states have adopted Common Core State Standards (CCSS) for
public schools, establishing new guidelines for student learning that are internationally
competitive. The CCSS represent a very different approach to teaching, learning, and
assessment—one focusing on fewer but more rigorous standards, and fostering a deeper
understanding of critical concepts and the practical applications of knowledge. Developed
over many years, tested, and proven to be effective, these new learning standards draw on
research on how students learn and how best to prepare them for college and the
increasingly competitive job market.
The purpose of the Mathematics Assessment Framework is to inform educators, test
developers, the public, and policy makers of the topics in the CCSS for math that will be
emphasized over the next two years in New Mexico's Standards Based Assessment (SBA).
This “bridge” assessment will be designed to incrementally shift the New Mexico SBA to a
closer alignment with the CCSS.
To spur its implementation of the Common Core State Standards, New Mexico is a member
of the Partnership for Assessment of Readiness for College and Careers (PARCC)—a
consortium of 24 states working together to develop a common assessment system for the
CCSS. New Mexico is a Governing State, or leader, in the PARCC consortium. The PARCC
assessments, which the consortium expects to be available to states in the 2014–2015
school year, will be designed to measure higher-order skills such as critical thinking,
reasoning, communications, and problem solving that are essential to college and career
readiness. In addition, these next generation assessments will help to determine if students
are progressing in their mastery of the content and skills tapped by the CCSS in English
language arts and mathematics. They will be designed, as well, to provide parents,
educators, and policymakers with comparable performance data on students’ proficiency
within and across states.
By 2015, all statewide accountability tests in New Mexico will be developed and
administered by PARCC. In other words, New Mexico students will be taking the same
standards based assessments as students in 23 other states. These tests will be delivered
and taken via computer.
New Mexico is seeking to better prepare teachers and students to meet the heightened
expectations of the new learning standards and the PARCC assessments by steadily raising
the alignment of the State’s standards based assessment to the CCSS. In school year 2012–
2013, for example, the SBA will test students on the New Mexico state standards for grades
4–8, 10, and 11, and there will be a Bridge Assessment for grade 3 that is dually aligned to
CCSS and the New Mexico state standards. This means that for third graders the SBA will
immediately begin to emphasize the learning goals of CCSS. In 2013–2014, there will be a
“bridge” SBA for students in all tested grades (3–8, 10, and 11) that will look and feel more
like the PARCC assessment. This means that students will experience test questions more
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
1
directly aligned to the CCSS. By 2014–2015, all New Mexican students will be taking the
PARCC assessments.
The New Mexico Public Education Department (PED) is responsible for developing
assessment frameworks for statewide testing during the transition to PARCC. These
assessment frameworks define in general terms the knowledge, skills, and abilities to be
assessed at each grade and describe the format of the tests and required achievement
levels. The assessment frameworks serve another important purpose as well. They signal a
shift towards greater transparency about the form of the SBA, emulating the openness
about test items and assessment support for teachers that will be implemented with the
PARCC system. In CCSS, teachers will become more expert about assessment, its purposes
and methodologies, and will be modeling in their own classrooms the examination
standards and protocols that are relied on at the state level.
II. About the Common Core State Standards
According to the expert groups who developed the CCSS for math, the Standards define
what students should understand and be able to do in their study of mathematics. Indeed,
they set grade-specific standards and benchmarks, but they do not dictate a curriculum or
specify teaching methods. This new generation of learning standards emphasizes both
content knowledge and conceptual understanding and highlights mathematical practices
related to problem solving, reasoning and proof, communication, representation, and
connections.
Organization
In the Common Core, mathematics follows a three-tiered structure that describes for each
grade as follows:
 Domains, which represent the ultimate skillsets that students must acquire
 Clusters, which are groups of related standards needed to perform complex tasks
 Standards that define what students must understand and be able to do
The CCSS for mathematics for grades 3–5 are organized into the following domains:
 Operations and Algebraic Thinking
 Number and Operations in Base Ten
 Number and Operations-Fractions
 Measurement and Data
 Geometry
The domains for grades 6–8 are these:
 Ratios and Proportional Relationships
 The Number System
 Expressions and Equation
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Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department


Geometry
Statistics and Probability
Functions are introduced as a domain in grade 8.
Because math skills are interconnected, standards within different domains and clusters
may be closely related. The grade-specific standards progress in complexity and rigor as
they move upward in grade level.
Mathematical Practices that Affect all Grades
The CCSS describe the characteristics of a student who meets the standards. They call these
characteristics the Standards for Mathematical Practice. These are the essential skills that
must be taught in every course and at every grade and against which students will be
assessed. These mathematical practices should be woven throughout the curriculum, and
invoked particularly where the standards indicate that students must “understand” a topic.
Standards for Mathematical Practice
 Make sense of problems and persevere in solving them
 Reason abstractly and quantitatively
 Construct viable arguments and critique the reasoning of others
 Model with mathematics
 Use appropriate tools strategically
 Attend to precision
 Look for and make use of structure
 Look for and express regularity in repeated reasoning
Learning Progressions
The CCSS were deliberately designed to improve teaching and learning in U.S. public schools
by recognizing how student learning progresses. The developers of the CCSS understood
that if the newly revised learning standards failed to lead to the alignment of coursework
within and across grades they would not promote learning. Learning progressions serve to
describe successively, incrementally more sophisticated ways of thinking within an
academic domain—showing how topics and concepts follow one another as students learn.
They lay out in words and examples what it means to move forward to more expert
understanding of a subject, and provide a picture of what it means to “improve” or “grow”
in academic proficiency. According to contemporary learning experts, “these pathways or
progressions serve to ground curriculum, instruction and assessment” (Heritage, 2008). For
this to happen, teachers and students alike need an understanding of how learning
develops.
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
3
The learning progressions, as articulated in the CCSS, provide guidance about the types of
assessment tasks that elicit evidence to support inferences about student achievement at
different points along the progression trajectory and vertically across grade levels. The
CCSS and the PARCC content frameworks move beyond the traditional horizontal (grade
level) definitions of standards to a vertical view of learning in which there is a sequence
along which students can move incrementally from novice to more advanced performance.
Thus the CCSS and this assessment framework describe what it means for students to move
over time toward a more expert understanding of the big ideas that bring coherence to the
academic domain of math.
Six “Shifts” in Mathematics
For New Mexico’s students to successfully master the CCSS, it is necessary for teachers to
realize the major differences between New Mexico’s current standards and those in the
CCSS, and how those differences will be reflected in the SBA. These six shifts are described
below.
Focus. The CCSS for mathematics increase focus on a more limited number of topics. The
purpose for this shift is to provide students with strong foundational knowledge and deep
conceptual understanding so they can transfer mathematical skills across concepts and
grade levels.
Coherence. The shift towards greater coherence means that topics are carefully connected
within and across grade levels so that students build greater understanding of a topic each
year, rather than visiting the topic in one section in one grade and moving on. For example,
concepts of multiplication and fractions are imbedded in almost every grade level so that
students continue to build on previous learning.
Fluency. Students need to develop speed and accuracy with simple calculations so they can
learn more advanced topics without being held back by a lack of fluency. This means
teachers will need to devote time to helping students memorize basic multiplication and
addition, for example, by teaching students memorization techniques and drilling students
in fundamental skills.
Deep Understanding. Teachers must begin to place less emphasis on “getting the answer”
and more emphasis on the multiple correct ways to solve a problem. Developing deep
understanding of a topic means that students will understand that there are many correct
ways to approach a mathematical problem, and students will be able to write and speak
about their problem-solving process.
Applications. Students will be expected to apply mathematics to “real-world” situations,
and use the appropriate mathematical tools to do so. Teachers in other disciplines and at
all grade levels can contribute by helping students use math to solve problems outside of
the mathematics classroom.
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Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
Dual Intensity. Students need to be able to both practice and understand with intensity.
For example, drilling with intensity can help students practice essential skills they need to
perform fluently. Regularly (intensely) describing problem-solving methods will help
students increase their understanding of a topic.
These changes mean that in the CCSS, assessments, including the Bridge Assessments, will
include problem solving that requires more intensive focus, the application of mathematical
concepts in different contexts, and mathematical modeling.
College and Career Readiness
College and career readiness for all students is another seminal idea in the CCSS. The
commitment to ensure that all public school students have the knowledge, skills and
abilities to be successful in post-secondary education marks a dramatic shift from most
states’ prior learning standards—many of which were little more than a disconnected,
‘laundry list’ description of grade-by-grade achievement. Using college and career
readiness as a benchmark also brings focus, coherence, and rigor to the CCSS.
New Mexico, like the other states adopting the CCSS, has agreed to further align
expectations for elementary and secondary school students with college and career
readiness—requiring all students to take challenging courses in high school that prepare
them for college. As a member of PARCC, New Mexico is streamlining its standards based
assessment system to hold students, teachers and schools accountable for “clearer, deeper,
stronger” learning standards, those that connect solidly with the knowledge and skills
needed to succeed in college and at work. In selecting items for the Bridge Assessments,
New Mexico will follow this same logic.
III. Design Principles for the Bridge Assessments
What is the Bridge Assessment Framework? Why is it Needed?
An assessment framework, unlike the highly detailed, technical blueprints that states use to
develop actual tests, lays out in plain language the design priorities of a standards based
assessment so that they can be understood by educational stakeholders and the public at
large. New Mexico’s Bridge Assessments for CCSS will be implemented starting in 2013. It
is the intent of the Public Education Department to describe the content, knowledge, and
skills to be tested across all public schools as New Mexico moves incrementally to
implement the CCSS in the next few years. By defining clearly and carefully what it is the
Bridge Assessments intend to measure, the Bridge Assessment framework offers a starting
place for a public conversation about what will be tested once the CCSS implemented in
2015.
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
5
The Bridge Assessment frameworks in English Language Arts (ELA) and mathematics will
explain to students, their parents, and teachers what we expect students to know and to be
able to do as the schools prepare students to be career and college ready. The frameworks
will be used to guide school-based instructional design teams as they identify the
complimentary sets of knowledge and skills needed for college and career readiness.
Design Goals
The 2013 and 2014 bridge standards based assessments in mathematics will include items
that measure the full range of student performance to be measured, including the
performances across the spectrum of high and low performing students. Like the current
assessments, the Bridge Assessments will provide data to inform instruction, including
measures of growth, and outline innovative approaches to assessment design. For these
reasons, this assessment framework ought to guide the professional development initiatives
for teachers of mathematics by identifying clearly the instructional shifts in content
knowledge and practices—the demands for rigor, fluency, and deep understanding—that
are the focus of the CCSS.
The assessment frameworks meet these goals by specifying student achievement levels,
and by describing the levels of thinking and reasoning that the Bridge Assessment items and
(or) tasks demand of the students; how those test items and tasks are distributed
proportionately across the content areas tested; and the various item and task formats to
be used, including multiple-choice and constructed response tasks. These technical details
are defined in the next section.
The Bridge Assessment frameworks build from the PARCC Model Content Frameworks in
math. In doing so, this assessment framework provides a sufficient level of specification to
guide the test development process and serve as a preliminary blueprint for constructing
the Bridge Assessment test forms in math now and in the future. At the same time, the PED
has set guidelines for establishing priority areas of assessment, including essential learning
standards as described by CCSS/PARCC, those most important to student development, and
those not well covered by current New Mexico standards. Finally, the Bridge Assessment
framework—though admittedly an interim document—ought to be durable enough to
remain relatively stable for the next three to five years.
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Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
Impact on Instruction: New Priorities for Teachers
After high school graduation, students’ abilities in the mathematical practices are at least as
important, if not more so, than the content they know. For that reason, the CCSS tend to
focus more on processes rather than just content. Future teaching, learning, and
assessment should therefore all become more process-focused. This can be accomplished
by infusing more performance tasks and authentic assessments into curricula. At the same
time, in CCSS teachers will find that assessments continue to be directly linked to curriculum
and instruction—in other words, the material in the standards is both what must be taught
and what will be assessed.
When designing assignments, teachers should keep in mind the over-arching goals of the
CCSS for math, which are to constantly integrate the mathematical practices throughout
students’ learning and help students become life-long critical thinkers and problem solvers.
These goals should be attained through an increased focus on algebraic or “abstract”
thinking, fluency in number and operations and greater understanding of proportional
relationships and probability.
The CCSS value depth of knowledge over breadth of knowledge. This means that New
Mexico teachers should expect to teach less content but with more complexity than they
may have in the past. In mathematics in particular, students will be learning topics at
earlier grades than before, which may challenge teachers to become fluent in mathematics
topics they are not currently teaching.
Another important implication of the CCSS for instruction is the need for teacher
collaboration. The learning progressions embedded in the CCSS for math require that
teachers collaborate vertically, to ensure that understanding and fluency build as students
move from grade-to-grade. Horizontal collaboration is also extremely important,
particularly due to the focus in CCSS for math on coherence across standards and promoting
deep understanding of core concepts. Teachers will need to develop their pedagogical skills
to support students’ ability to access core math concepts from a variety of perspectives.
Finally, experts acknowledge that when implementing new standards, pedagogical changes
nearly always lag change in the assessments. New Mexico’s teachers must begin to change
instructional techniques and material now, in anticipation of the Bridge Assessments and
the full transition to CCSS. Although the bridge standards based assessments will not be as
much of a change as the PARCC exams, they will reflect the CCSS, requiring students to
demonstrate a deeper knowledge of topics. Teachers can anticipate the need for:
 More fidelity to the standards in classroom teaching
 More professional development to train teachers in the CCSS and inform them of
the how the CCSS will be assessed
 Increased emphasis on collaborative lesson planning within and across grades
 New strategies for classroom management and use of time in the classroom
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
7
Use of Technology
The Bridge Assessment for third graders in 2013 will be a paper and pencil test. The Bridge
Assessments in 2014 and beyond should be developed to be delivered mostly by computer,
utilizing formats and technology, like those for short-cycle testing, already available in the
districts.
IV. Reporting and Accountability
Policy and Definitions
Technology
The PARCC assessments are intended to become computer-based, statewide assessments.
There are many benefits to computer-based assessment, including the following:
 The ability to make the assessments adaptive increases the opportunity to better
test students at the lower and higher ends of achievement. However, the PARCC
assessment will not be adaptive.
 Scoring can be accomplished and reported more quickly than on paper and pencil
tests.
 Standardized scaffolds can be imbedded in the program (see “Scaffolding” below).
However, issues of equity would need to be addressed in implementing statewide
computer-based assessment including the following:
 Disparity among districts that cannot afford enough computers for all students to
take the exams
 Differences in students' experiences using technology because of lack of access
 Differences in access to broadband internet in different areas of the state (if the
assessment is web-based)
Scaffolding
The idea of providing students with “scaffolding”—the additional materials and aids
available to students during an assessment—is evolving, and will play a greater role in
assessments as technology advances. More often than not, students use “scaffolds” in
order to solve more complex problems than they could without such aids. The inclusion of
scaffolded assessment items and tasks allows students to demonstrate what they know and
can do without placing undo burden on test-taking strategies that emphasize rote
memorization or simply recognizing or choosing an answer from a number (multiple) of
choices. Examples of these types of assessment items would include those requiring
students to use a graphing calculator or a number-line graphic on mathematics
assessments. With the implementation of the CCSS comes the need to assess more
complex topics at a deeper level of conceptual understanding and to tap into students’
higher-order reasoning skills, much of which will require the use of scaffolded assessment
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Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
items not previously available to students in current testing programs.
The shift towards computer-based testing changes the types of scaffolds or supports
available to students and allows test designers to imbed standardized scaffolds within the
assessment itself. However, before New Mexico transitions toward computer-based
assessments, the challenge of providing students with equitable and standardized scaffolds
statewide will continue. For example, if certain tools are viewed as legitimate scaffolds and
incorporated into the bridge assessments, the question arises whether they should be
provided by the PED, the schools, or if students should be required to bring their own.
Complexity
Increased complexity in standards based assessments is intended to better measure depth
of knowledge. Multi-part test items on one problem are intended to provide a rich picture
of students' understanding, but to serve that purpose they must be properly constructed.
According to the National Assessment of Educational Progress (NAEP)
 Low-complexity items expect students to recall or recognize concepts or procedures;
 Moderate-complexity items involve more flexibility of thinking and choice among
alternatives than do those in the low-complexity category; and
 High-complexity items make heavy demands on students, because they are
expected to use reasoning, planning, analysis, judgment, and creative thought.
Item Formats
The State’s testing provider, following the guidelines in this document, will develop test
items and tasks aligned with CCSS for use in the 2014 Bridge Assessment. Those items will
be field tested in the spring of 2013 as part of the SBA, and will be scored but will not be
counted toward school accountability grades. Field-tested items will be selected for
inclusion in the 2014 Bridge Assessment. These items too will be scored but not count
toward school accountability grades.
The distribution of items on the “bridge” SBAs will remain largely consistent with that used
by the SBA over the past two years, with about 80 percent of items being multiple choice,
12 percent short answer, and 8 percent extended response. Constructed response items
carry greater weight in the scoring of the SBA, so that multiple choice answers account for
about 60 percent of the total score, and constructed response items account for 40 percent.
These distribution and scoring percentages represent the state’s targets, and are not meant
to be construed as formulas for the Bridge Assessments in 2013 or beyond.
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
9
Achievement Levels
The Bridge Assessments will follow the state’s current proficiency ratings for reporting
scores based on New Mexico Academic Content Standards, as outlined in the New Mexico
Standards Based Assessment: Standard Setting Report (Measured Progress, 2011) as
follows:
 Beginning Step
 Nearing Proficiency
 Proficient
 Advanced
In order to preserve the integrity of trend data on student performance, only those items
that are aligned with the New Mexico Academic Content Standards and/or dually aligned
with the New Mexico Academic Content Standards and the CCSS will be used in calculating
student proficiency.
SBA Results and the A-F Accountability System
School grades in the A-F accountability system will continue to be based on growth in
student scores using New Mexico Academic Content Standards. The items and tasks on the
bridge standards based assessments that will be scored for accountability purposes will be
aligned with current New Mexico state standards. Some of the SBA’s items and tasks also
align with CCSS, particularly in Reading, and they will be scored for accountability purposes.
This methodology ensures that the performance data the state is using to compute school
grades is reliable and valid.
The Bridge Assessment will include newly developed items aligned with CCSS starting in
2014. Some of those items will measure essential skills not in New Mexico Academic
Content Standards. The PED will issue a separate report to districts about baseline
performance on CCSS items and tasks.
V. Critical Skills
In New Mexico and elsewhere, past standards based assessments have measured student
proficiency by taking snapshots of student performance on individual grade-level standards.
In the Common Core, New Mexico’s assessments are being designed to provide a more
complete picture. The Bridge Assessments, modeling the PARCC assessments, follow, and
to some extent predict, whether classroom instruction is leading students toward being well
prepared for post-secondary education by the time of high school graduation.
The CCSS for math describe critical areas within each grade level. Because not all content is
equally important, the CCSS for math place more emphasis on certain clusters. This means
that some clusters require more attention based on the depth of their ideas, the time that
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Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
they take to master, and/or their importance to future mathematics or the demands of
college and career readiness. Deep understanding of these critical skills is accomplished
through the Standards for Mathematical Practice and the concept of fluency, which
indicates that a student can carry out procedures accurately, flexibly, efficiently, and
appropriately.
The presence of these critical areas within the standards should not imply that clusters not
listed as critical areas are not important (see, for example, the detailed information
provided in Tables 1 and 2 in the Appendix). Neglecting materials in the standards will leave
gaps in student skill and understanding and may leave students unprepared for the
challenges of a later grade. All standards figure in a mathematical education, and will
therefore be eligible for inclusion on the PARCC assessment. However, among all of the
CCSS, we believe that the critical skills identified here should be given priority in instruction
and assessment over the next several years.
The following high-level description of essential skillsets synthesizes the detailed
descriptions of what students must understand and be able to do as found in the domains,
clusters and standards. The skillsets have been identified as critical for one or more of the
following reasons:
 They are fundamental building blocks in college and career readiness.
 They strongly contribute to the development of the Mathematical Practices and
have been identified as critical by the CCSS themselves.
 They are areas not adequately covered by New Mexico’s current standards.
Grades 3–5
In grades three through five, students are developing the skills that lay the foundation for
all subsequent work in mathematics, are essential to performing college- and career-related
mathematical skills with fluency, and help them understand the world around them. The
skills described below are the most basic abilities that students need throughout their lives
in personal finances, cooking, and home maintenance. Without these skills, students
cannot succeed in higher-level mathematics or entry-level jobs in almost every career field.
For example, if students cannot add, subtract, multiply, and divide multi-digit numbers and
decimals fluently without a calculator, they are unlikely to succeed in almost any job they
may enter during or after high school.
The Mathematics Learning Progressions: Grades 3–5 table (Table 1) found in the Appendix
provides more detail on what students need to know and what they should be tested on in
each grade.
Operations and Algebraic Thinking
In grade three, the CCSS require that students be able to multiply and divide numbers up to
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
11
100. In so doing, students must have an understanding of the properties of multiplication
and the relationship between multiplication and division. In grade 4, this skill should be
further developed into an understanding of and ability to perform multi-digit multiplication
and division fluently. Fourth grade students should become familiar with factors and
multiples, and should be able to analyze patterns in numbers. Mastery of these skills is vital
to performing higher-level mathematics efficiently.
In fifth grade, students should be able to write and interpret numerical expressions and
analyze patterns and relationships. These abilities help develop the logical reasoning skills
that students need in many subjects throughout their education.
Number and Operations in Base Ten
Third and Fourth grade students should develop an understanding of digit place value (e.g.,
ones, tens, thousands) in order to add, subtract, multiply and divide multi-digit numbers. In
grade five, students should be adding, subtracting, multiplying and dividing multi-digit
whole numbers and decimals to the hundredths place fluently. These basic skills are
necessary for all subsequent work in mathematics, and are particularly important to the
development of life skills, such as calculations with money.
Numbers and Operations—Fractions
In grade three, students should develop an understanding of fractions as numbers. In 4th
grade, students’ understanding of fractions should have evolved to adding and subtracting
fractions with the same denominators (1/4 + 3/4, 3/8 + 5/8, etc.) and multiplying fractions
by whole numbers. Fifth grade students’ skills with fractions should be extended to fluency
adding and subtracting fractions with different numerators and denominators, and building
on what they learned in 4th grade to divide fractions. Not only does a fluent understanding
of fractions lay the foundation for proportional reasoning, but it is necessary for life skills
like calculating measurements for cooking.
Measurement and Data
The major emphasis in Measurement and Data in grades three through five is on perimeter,
area and volume. In grade three, students need to understand the concept of area and how
it relates to multiplication and division. Third graders also need to recognize perimeter and
tell the difference between area and perimeter and their calculations. In grade four,
students learn about conversion of units and how they relate to perimeter, area, and
volume. Fourth graders also need to learn about the measurements of angles in geometry.
In fifth grade, students’ understanding of measurement and data extends to calculations of
volume and understanding how these calculations relate to multiplication and division.
Understanding perimeter, area, and volume and their calculations is vital in later
mathematics from high school geometry to calculus, and in everyday life these concepts will
help them understand the food and home products they are purchasing as consumers.
12
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
Geometry
In third grade students should understand different types of shapes and their qualities,
including how different geometric shapes relate to each other. In grade four, students’
abilities should be further developed to understand that geometric figures can be analyzed
and classified based on their properties such as parallel or perpendicular sides, different
angle measures, and symmetric figures. Fifth graders should be able to classify twodimensional shapes based on their characteristics. Also in fifth grade students should be
introduced to graphing points on the coordinate plane. These subjects develop spatial
reasoning skills that are necessary not only in higher mathematics but in life. Understanding
shapes has applications in many professions and areas of life including construction and
architecture, visual arts, and sewing.
Grades 6–8
The intensive application of the Mathematical Practices and material learned in grades 6
through 8 relate directly to the development of skills that students need to be ready for
college and careers. Most importantly, students will further develop the mathematical and
logical reasoning abilities they will need in other academic subjects and throughout life.
The Mathematics Learning Progressions: Grades 6–8 table (Table 2) found in the Appendix
provides more detail on what students need to know and what they should be tested on in
each grade.
Ratios and Proportional Relationships
In sixth grade students are introduced to the concepts of ratio and proportion and should
use ratios to solve problems. In seventh grade students need to extend this understanding
analyze and use proportional relationships in problem solving. Sixth and seventh graders
should be able to use ratio reasoning to solve problems by viewing equivalent ratios and
rates; analyzing simple drawings that indicate relative sized of quantities; and connecting
understanding of multiplication and division of ratios and rates.
Although ratios and proportional relationships are not addressed so specifically in 8 th grade,
these skills prepare students to make connections between proportional relationships,
lines, and linear equations. These skills also prepare students to learn about financing,
interest rates, payments, and discounts—all necessary topics for interpreting financial
information in the workplace and for making financial decisions throughout their lives.
The Number System
In 6th grade, students need to extend previous understanding to multiply and divide
fractions with fractions. Sixth graders also need to be able to find factors and multiples of
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
13
multi-digit numbers, and apply previous learning to rational numbers. In seventh grade
students should build on these skills to perform operations with rational numbers. Eighth
graders should learn that there are numbers that exist that are not rational, and how to
approximate them with rational numbers. These abilities are essential in algebra and
higher-level mathematics. Furthermore they help students understand that the number
system extends beyond previous conceptions of numbers.
Expressions and Equations
Students in sixth grade will begin to extend what they learned about arithmetic to algebraic
expressions. They should learn about and solve one-variable equations and analyze
relationships between dependent and independent variables. In seventh grade, students
will be expected to create expressions and solve problems using numerical and algebraic
expressions and equations. Eighth graders will build on their previous learning to solve
linear equations and pairs of linear equations fluently. They also should be working with
radical numbers and integer exponents. These are foundational skills for college readiness,
and the reasoning and problem solving abilities learned by mastering these topics are
required in many 21st century careers.
Geometry
Sixth grade students should be able to solve mathematical problems involving area, surface
area, and volume. In seventh grade these problems should be extended to include angle
measure, and students should be able to construct geometric figures and describe
relationships between them. Eighth graders will be expected to build on these skills by
solving problems involving the volume of cylinders, cones, and spheres, which not only
improves spatial reasoning but is essential in higher mathematics in high school. Students
in eighth grade should also understand the Pythagorean Theorem, which is essential for
understanding right triangle geometry, which in turn leads to trigonometry in high school.
Statistics and Probability
It is necessary for students in middle school to begin learning to explore critique and
understand the world around them. Understanding statistics and probability is essential for
interpreting data and information they may encounter everywhere from the internet to the
doctor’s office. Sixth graders will begin to develop an understanding of statistical variability
and be able to summarize and describe distributions. In seventh grade, students extend
these skills to drawing inferences about a population and compare two populations.
Seventh graders should also be able to develop, use, and evaluate probability models. In
eighth grade, students should build on earlier skills to patterns between two quantities,
such as with a scatterplot, and they should learn that lines may be used to model
relationships between two variables.
14
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
Functions (Grade 8)
Functions are introduced in grade 8. Students will be expected to be able to define,
evaluate, and compare functions, and use functions to model relationships between
quantities. Developing an understanding of functions before high school will be critical to
students’ understanding of algebra. Functions are used in many subject areas that students
may become interested in later in life in college: functions are used to model real-life
situations in subjects including (but not at all limited to) finance, economics, statistics,
biology, physics, chemistry, and engineering. Beginning to understand functions in 8 th grade
is truly essential to becoming college- and career-ready.
VI. Conclusion
The CCSS set out clearly the content knowledge and skills in mathematics needed to
succeed in college and the world of work. The assessment framework presented in this
document is intended to set forward-thinking goals for students, their parents and their
teachers. If successful, the Bridge Assessment built from this framework will begin to set
the stage for New Mexico public education not only beginning in 2013, but beyond and for
the next decade. The CCSS and the State’s standards based assessments will raise the bar
for all New Mexico’s students. Neither the CCSS nor the bridge assessments are intended to
dictate curriculum, pedagogy, or the delivery of instruction. The school districts across New
Mexico are expected to handle the transition to the CCSS in different ways. This assessment
framework is intended to help guide this transition.
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
15
VII. Appendices
Table 1: Mathematics Learning Progressions: Grades 3–5
Mathematics 3–5 Grade Level Cluster Progressions
16
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
Expressions and Equations
The Number System
Ratios and
Proportional
Relationships
Table 2: Mathematics Learning Progressions: Grades 6–8
Mathematics
Grade
Level
Cluster
Progressions
Mathematics
6–86-8
Grade
Level
Cluster
Progressions
Grade 6
Grade 7
Understand ratio concepts and use ratio
reasoning to solve problems by viewing
equivalent ratios and rates, analyzing simple
drawings that indicate relative size of quantities,
and connecting their understanding of
multiplication and division of ratios and rates.
Analyze proportional relationships and use
them to solve real-world and mathematical
problems.
Apply and extend, with fluency, previous
understandings of multiplication and division to
divide fractions by fractions, and understand
and explain why the procedures make sense.
Compute fluently with multi-digit numbers and
find common factors and multiples.
Apply and extend previous understandings of
numbers to the system of rational numbers
including absolute value and location of points
in all four quadrants of the coordinate plane.
*Apply and extend previous understandings of
arithmetic to algebraic expressions in order to
solve problems and identify when two
expressions are equivalent.
*Reason about and solve one-variable
equations and inequalities.
Grade 8
Apply and extend with fluency, previous
understandings of operations with fractions
to add, subtract, multiply, and divide all
rational numbers.
Use properties of operations to generate
equivalent expressions.
Know that there are numbers that are not
rational and approximate them by rational
numbers.
*Solve real-life and mathematical problems
using numerical and algebraic expressions
and formulate expressions and equations in
one variable to solve equations.
Understand the connections between
proportional relationships, lines, and
linear equations.
Represent and analyze quantitative
relationships between dependent and
independent variables.
Analyze and solve linear equations and
pairs of simultaneous linear equations
fluently.
Work with radicals and integer exponents.
*Use functions to model relationships
between quantities.
Geometry
*Solve real-world and mathematical problems
involving area, surface area, and volume.
*Solve real-life and mathematical problems
involving angle measure, area, surface area,
and volume.
Solve real-world and mathematical
problems involving volume of cylinders,
cones, and spheres with fluency.
Draw, construct and describe geometrical
figures and describe the relationships
between them.
Understand congruence and similarity
using physical models, transparencies, or
geometry software.
Functions
Statistics and
Probability
Understand and apply the Pythagorean
Theorem.
Develop an understanding of statistical
variability including utilizing different ways to
measure center.
Summarize and describe distributions.
Draw informal comparative inferences
about two populations
Use random sampling to draw inferences
about a population.
*Investigate chance processes and develop,
use, and evaluate probability models.
*Investigate patterns of association in
bivariate data.
*Define, evaluate, and compare
properties of two functions each
represented in a different way.
*Use functions to model relationships
between quantities.
Shaded Cells = Critical Focus Areas for CCSS
*Standards with an asterisk appear in the New Mexico Academic Content Standards at a different grade level than in the CCSS
Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department
17
Bibliography
Common Core State Standards for Mathematics, Common Core State Standards Initiative,
Council of Chief State Schools Officers and National Governors Association, Washington, DC.
Correspondence Between the New Mexico Content Standards and the Common Core State
Standards for English Language Arts and Mathematics: Summary Report, WestEd, San
Francisco, California, December 19, 2011.
Illinois Mathematics Assessment Framework Grades 3-8: State Assessments Beginning
Spring 2006, Illinois State Board of Education, Chicago, Illinois, September 2004.
Mathematics Framework for the 2011 National Assessment of Educational Progress,
National Assessment Governing Board, Washington, DC, September 2010.
New Mexico Math Assessment Priorities, New Mexico Public Education Department, Santa
Fe, New Mexico, 2012.
Suggested Outline for Mathematics and English Language Arts Frameworks Grades 6–12,
Howard T. Everson, The College Board, New York, New York, (unpublished draft)
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Math Assessment Frameworks, Grades 3–8, New Mexico Public Education Department