2. Purpose of Training Module
• Introduce primary source materials
– Common Core State Standards for Mathematics
– Smarter Balanced Mathematics Content Specifications
– Smarter Balanced Mathematics Item Specifications
• Describe how these documents will be used
for item writing
• Define Depth of Knowledge
3. Foundation for Item and Task Development
Common Core
State Standards
Smarter Balanced
Content Specifications
Smarter Balanced Item
and Task Specifications
Items and
Performance Tasks
4. Structure of the
Common Core State Standards for Mathematics
• Research-based learning progressions
• Internationally benchmarked
• Focused and coherent
• Standards for Mathematical Practice
– Identify important processes and proficiencies
• Standards for Mathematical Content
– Grade specific expectations
5. Structure of the
Common Core State Standards for Mathematics
STANDARD
CLUSTER
DOMAIN Number and Operations in Base Ten 3.NBT
Use place value understanding and properties of operations to
perform multi-digit arithmetic.
1. Use place value understanding to round whole numbers to the nearest
10 or 100.
2. Fluently add and subtract within 1000 using strategies and algorithms
based on place value, properties of operations, and/or the relationship
between addition and subtraction.
3. Multiply one-digit whole numbers by multiples of 10 in the range
10-90 (e.g., 9 x 80, 5 x 60) using strategies on place and properties of
operations..
6. Structure of the Smarter Balanced Assessment
Consortium Mathematics Content Specifications
• Introduction and Background
• Claims and Assessment Targets
• Rationale and Evidence for Each Claim
• Cognitive Rigor Matrix / Depth of Knowledge
• General Considerations
7. Cognitive Rigor and Depth of Knowledge
• The level of complexity of the cognitive demand.
– Level 1: Recall and Reproduction
• Requires eliciting information such as a fact, definition, term,
or a simple procedure, as well as performing a simple algorithm
or applying a formula.
– Level 2: Basic Skills and Concepts
• Requires the engagement of some mental processing beyond
a recall of information.
– Level 3: Strategic Thinking and Reasoning
• Requires reasoning, planning, using evidence, and explanations
of thinking.
– Level 4: Extended Thinking
• Requires complex reasoning, planning, developing, and
thinking most likely over an extended period of time.
8. Level 1 Example
Grade 8
Select all of the expressions that have a value between 0 and 1.
87 8–12
74
7–3
1
3
2
1
3
9
(–5)6
(–5)10
9. Level 2 Example
Grade 8
A cylindrical tank has a height of 10 feet and
a radius of 4 feet. Jane fills this tank with water
at a rate of 8 cubic feet per minute. How many
minutes will it take Jane to completely fill the
tank without overflowing at this rate?
Round your answer to the nearest minute.
10. Level 3 Example
Grade 8
The total cost for an order of shirts from a company consists of the cost for
each shirt plus a one-time design fee. The cost for each shirt is the same
no matter how many shirts are ordered.
The company provides the following examples to customers to help them
estimate the total cost for an order of shirts.
• 50 shirts cost $349.50
• 500 shirts cost $2370
Part A: Using the examples provided, what is the cost for each shirt, not
including the one-time design fee? Explain how you found your answer.
Part B: What is the cost of the one-time design fee? Explain how you found
your answer.
11. Level 4 Example
Grade 8
During the task, the student assumes the role of an architect
who is responsible for designing the best plan for a park with
area and financial restraints. The student completes tasks in
which he/she compares the costs of different bids, determines
what facilities should be given priority in the park, and then
develops a scale drawing of the best design for the park and an
explanation of the choices made. This investigation is done in
class using a calculator, an applet to construct the scale drawing,
and a spreadsheet.
12. Cognitive Rigor Matrix
This matrix from the Smarter Balanced Content Specifications for Mathematics draws from both Bloom’s
(revised) Taxonomy of Educational Objectives and Webb’s Depth-of-Knowledge Levels below.
14. Mathematics Assessment Claims
• Claim 1: Concepts and Procedures
– Students can explain and apply mathematical concepts and interpret
and carry out mathematical procedures with precision and fluency
• Claim 2: Problem Solving
– Students can solve a range of complex well-posed problems in pure
and applied mathematics, making productive use of knowledge and
problem solving strategies
• Claim 3: Communicating Reasoning
– Students can clearly and precisely construct viable arguments to
support their own reasoning and to critique the reasoning of others
• Claim 4: Modeling and Data Analysis
– Students can analyze complex, real-world scenarios and can construct
and use mathematical models to interpret and solve problems
15. Claim 1
Concepts and Procedures
Students can explain and apply mathematical concepts
and interpret and carry out mathematical procedures with
precision and fluency.
Grade Level
Number of
Assessment Targets
3 11
4 12
5 11
6 10
7 9
8 10
11 16
16. Claim 1
Concepts and Procedures
Grade 4
Operations and Algebraic Thinking
Target A [m]: Use the four operations with whole numbers to solve problems. (DOK 1, 2)
Tasks for this target will require students to use the four operations to solve straightforward,
one-step contextual word problems in situations involving equal groups, arrays, and finding
an unknown number, including problems where the remainder must be interpreted. Some of
these tasks will draw on contexts in 4.MD Target I using measurement quantities such as time,
liquid volume, and masses/weights of objects, and money (with decimal representations
limited to those described in standards 4.NF.6 and 4.NF.7).
17. Claim 1 Assessment Targets
Grade 3
Operations and Algebraic Thinking
A. Represent and solve problems involving multiplication and
division.
B. Understand properties of multiplication and the relationship
between multiplication and division.
C. Multiply and divide within 100.
D. Solve problems involving the four operations, and identify and
explain patterns in arithmetic.
Number and Operations – Base 10
E. Use place value understanding and properties of arithmetic to
perform multi-digit arithmetic.
18. Claim 1 Assessment Targets
Grade 3
Numbers and Operations – Fractions
F. Develop understanding of fractions as numbers.
Measurement and Data
G. Solve problems involving measurement and estimation of intervals
of time, liquid volumes, and masses of objects.
H. Represent and interpret data.
I. Geometric measurement: understand concepts of area and relate
area to multiplication and to addition.
J. Geometric measurement: recognize perimeter as an attribute of
plane figures and distinguish between linear and area measures.
Geometry
K. Reason with shapes and their attributes.
19. Claim 1 Assessment Targets
Grade 4
Operations and Algebraic Thinking
A. Use the four operations with whole numbers to solve problems.
B. Gain familiarity with factors and multiples.
C. Generate and analyze patterns.
D. Generalize place value understanding for multi-digit whole
numbers.
E. Use place value understanding and properties of operations to
perform multi-digit arithmetic.
Number and Operations – Fractions
F. Extend understanding of fraction equivalence and ordering.
G. Build fractions from unit fractions by applying and extending
previous understandings of operations on whole numbers.
H. Understand decimal notation for fractions, and compare
decimal fractions.
20. Claim 1 Assessment Targets
Grade 4
Measurement and Data
I. Solve problems involving measurement and conversion of
measurements from a larger unit to a smaller unit.
J. Represent and interpret data.
K. Geometric measurement: understand concepts of angle and
measure angles.
Geometry
L. Draw and identify lines and angles, and classify shapes by
properties of their lines and angles.
21. Claim 1 Assessment Targets
Grade 5
Operations and Algebraic Thinking
A. Write and interpret numerical expressions.
B. Analyze patterns and relationships.
Number and Operations – Base 10
C. Understand the place value system.
D. Perform operations with multi-digit whole numbers and with
decimals to hundredths.
Number and Operations – Fractions
E. Use equivalent fractions as a strategy to add and subtract
fractions.
F. Apply and extend previous understandings of multiplication
and division to multiply and divide fractions.
22. Claim 1 Assessment Targets
Grade 5
Measurement and Data
G. Convert like measurement units within a given measurement
system.
H. Represent and interpret data.
I. Geometric measurement: understand concepts of volume and
relate volume to multiplication and to addition.
Geometry
J. Graph points on the coordinate plane to solve real-world and
mathematical problems.
K. Classify two-dimensional figures into categories based on their
properties.
23. Claim 1 Assessment Targets
Grade 6
Ratios and Proportional Relationships
A. Understand ratio concepts and use ratio reasoning to solve
problems.
The Number System
B. Apply and extend previous understandings of multiplication and
division to divide fractions by fractions.
C. Compute fluently with multi-digit numbers and find common
factors and multiples.
D. Apply and extend previous understandings of numbers to the
system of rational numbers.
24. Claim 1 Assessment Targets
Grade 6
Expressions and Equations
E. Apply and extend previous understandings of arithmetic to
algebraic expressions.
F. Reason about and solve one-variable equations and
inequalities.
G. Represent and analyze quantitative relationships between
dependent and independent variables.
Geometry
H. Solve real-world and mathematical problems involving area,
surface area, and volume.
Statistics and Probability
I. Develop understanding of statistical variability.
J. Summarize and describe distributions.
25. Claim 1 Assessment Targets
Grade 7
Ratios and Proportional Relationships
A. Analyze proportional relationships and use them to solve real-
world and mathematical problems.
The Number System
B. Apply and extend previous understandings of operations with
fractions to add, subtract, multiply, and divide rational numbers.
Expressions and Equations
C. Use properties of operations to generate equivalent expressions.
D. Solve real-life and mathematical problems using numerical and
algebraic expressions and equations.
26. Claim 1 Assessment Targets
Grade 7
Geometry
E. Draw, construct and describe geometrical figures and describe the
relationships between them.
F. Solve real-life and mathematical problems involving angle
measure, area, surface area, and volume.
Statistics and Probability
G. Use random sampling to draw inferences about a population.
H. Draw informal comparative inferences about two populations.
I. Investigate chance processes and develop, use, and evaluate
probability models.
27. Claim 1 Assessment Targets
Grade 8
The Number System
A. Know that there are numbers that are not rational, and
approximate them by rational numbers.
Expressions and Equations
B. Work with radicals and integer exponents.
C. Understand the connections between proportional relationships,
lines, and linear equations.
D. Analyze and solve linear equations and pairs of simultaneous linear
equations.
Functions
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
28. Claim 1 Assessment Targets
Grade 8
Geometry
G. Understand congruence and similarity using physical models,
transparencies, or geometry software.
H. Understand and apply the Pythagorean theorem.
I. Solve real-world and mathematical problems involving volume of
cylinders, cones and spheres.
Statistics and Probability
J. Investigate patterns of association in bivariate data.
29. Claim 1 Assessment Targets
Grade 11
Number and Quantity
A. Extend the properties of exponents to rational exponents.
B. Use properties of rational and irrational numbers.
C. Reason quantitatively and use units to solve problems.
Algebra
D. Interpret the structure of expressions.
E. Write expressions in equivalent forms to solve problems.
F. Perform arithmetic operations on polynomials.
G. Create equations that describe numbers or relationships.
H. Understand solving equations as a process of reasoning and
explain the reasoning.
I. Solve equations and inequalities in one variable.
J. Represent and solve equations and inequalities
graphically.
30. Claim 1 Assessment Targets
Grade 11
Functions
K. Understand the concept of a function and use function
notation.
L. Interpret functions that arise in applications in terms of a
context.
M. Analyze functions using different representations.
N. Build a function that models a relationship between two
quantities.
Geometry
O. Prove geometric theorems.
Statistics and Probability
P. Summarize, represent and interpret data on a single count or
measurement variable.
31. Claims 2, 3, and 4
• Assessment Targets for Claims
2, 3, and 4 are not divided into a
grade-by-grade description.
• A general set of assessment targets
applicable across grade levels.
32. Assessment Targets
Claim 2 – Problem Solving
A. Apply mathematics to solve well-posed problems arising in
everyday life, society, and the workplace
B. Select and use tools strategically
C. Interpret results in the context of the situation
D. Identify important quantities in a practical situation and
map their relationships.
Claim 2: Students can solve a range of complex well-posed problems in
pure and applied mathematics, making productive use of knowledge
and problem solving strategies.
33. Assessment Targets
Claim 3 – Communicating Reason
A. Test propositions or conjectures with specific examples.
B. Construct, autonomously, chains of reasoning that justify or refute
propositions or conjectures.
C. State logical assumptions being used.
D. Use the technique of breaking an argument into cases.
E. Distinguish correct logic or reasoning from that which is flawed,
and—if there is a flaw in the argument—explain what it is.
F. Base arguments on concrete referents such as objects, drawings,
diagrams, and actions.
G. Determine conditions under which an argument does and
does not apply.
Claim 3: Students can clearly and precisely construct viable
arguments to support their own reasoning and to critique the
reasoning of others.
34. Assessment Targets
Claim 4 – Modeling and Data Analysis
A. Apply mathematics to solve problems arising in everyday life, society,
and the workplace.
B. Construct, autonomously, chains of reasoning to justify mathematical
models used, interpretations made, and solutions proposed for a
complex problem.
C. State logical assumptions being used.
D. Interpret results in the context of a situation.
E. Analyze the adequacy of and make improvement to an existing model
or develop a mathematical model of a real phenomenon.
F. Identify important quantities in a practical situation and map their
relationships.
G. Identify, analyze, and synthesize relevant external resources to pose
or solve problems.
Claim 4: Students can analyze complex, real-world scenarios and
can construct and use mathematical models to interpret and
solve problems.
35. Structure of the Smarter Balanced
Assessment Consortium Mathematics
Item and Task Specifications
• Evidence Statements
• Task Models
• Sample Items
• Mathematics Glossary
37. Grade 5 Evidence Statement
Assessment Target C:
Understand the place value system. (DOK 1, 2)
Evidence Statement:
The student explains patterns in the number of
zeros of the product/quotient when
multiplying/dividing a number by powers of 10.
38. Grade 5 Task Model
Evidence Statement:
The student explains patterns
in the number of zeros of
the product/quotient when
multiplying/dividing a number
by powers of 10.
Task Model
2. CR (DOK 1, 2)
Prompt Features 1: The student is prompted
to explain patterns in the number of zeros of
a product when multiplying a number by
powers of 10.
Stimuli: The student is presented with a
multiplication problem involving powers of
10.
Prompt Features 2: The student is prompted
to explain patterns in the number of zeros of
a quotient when dividing a number by powers
of 10.
Stimuli: The student is presented with a
division problem involving powers of 10.
40. Module Summary
• Common Core State Standards: foundation for
claims and assessment targets
• Smarter Balanced Content Specifications: define
claims and assessment targets
– Depth of Knowledge: Level of cognitive processes
applied by students
• Smarter Balanced Item and Task Specifications:
define evidence required and describe task models
Editor's Notes
Welcome to the Smarter Balanced Assessment Consortium’s Mathematics Content Specifications, Item Specifications, and Depth of Knowledge Training Module.
This training module focuses on three main topics.
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First, this module provides an overview of the structure of the Common Core State Standards for Mathematics, the Smarter Balanced Mathematics Content Specifications, and the Smarter Balanced Mathematics Item Specifications.
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A description of how these documents are used by item writers to write high-quality test items and tasks is also provided.
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Finally, this module provides an introduction to the Depth of Knowledge taxonomy for cognitive complexity.
Let’s begin with an overview of the structure of each resource document.
As will be explored in greater detail in this module, the Common Core State Standards
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Provided the foundation for the Smarter Balanced Content Specifications.
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The content specifications were then used to inform the development of the Smarter Balanced Item Specifications.
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Item and task writers are expected to use all three documents to guide the development of Smarter Balanced items and tasks.
The Smarter Balanced Assessment Consortium’s Math Content Specifications and Item Specifications are based on the Common Core State Standards for Mathematics.
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The development of these standards began with research-based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time.
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The standards are internationally benchmarked
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and are more focused and coherent to address the problem of a mathematics curriculum that is “a mile wide and an inch deep”.
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The mathematics standards are comprised of The Standards for Mathematical Practice and the Standards for Mathematical Content.
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The Standards for Mathematical Practice identify the important processes and proficiencies that students should develop.
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The Standards for Mathematical Content set grade-specific expectations for what students should understand and be able to do.
The Common Core State Standards for Mathematics are organized into three levels;
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Standards, which define what students should understand and be able to do;
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Clusters, which are groups of related standards;
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And domains or conceptual categories, which are larger groups of related standards. Standards from different domains may sometimes be closely related.
The Smarter Balanced Assessment Consortium Mathematics Content Specifications describe the Consortium’s current specification of critically important claims about student learning that are derived from the Common Core State Standards for Mathematics. Item writers must write items and tasks that align to these claims and assessment targets.
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The introduction and background section presents a discussion of several issues that influenced the development of the content specifications. For example, the Smarter Balanced Consortium is utilizing evidence-centered design methodologies to create its assessment program.
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The general considerations section contains discussions of four topics: how to use various types of items and tasks to connect content and practice, focus and coherence in mathematics, judicious coverage of the Common Core State Standards for Mathematics, and strategic uses of technology.
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The four claims identified in the Mathematics Content Specifications serve as the foundation for the Item and Task Specifications. Each claim is a summary statement about the knowledge and skills students will be expected to demonstrate on the assessment related to a particular aspect of the Common Core State Standard for Mathematics. Each claim is accompanied by a set of assessment targets that provide more detail about the range of content and Depth of Knowledge levels.
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The assessment targets are intended to support the development of high-quality items and tasks that contribute evidence to the claims.
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The content specifications also include a discussion of cognitive rigor. The Smarter Balanced Consortium is using a cognitive rigor matrix that is based on Bloom’s revised Taxonomy of Educational Objectives and Webb’s Depth-of-Knowledge Levels.
Smarter Balanced items and tasks will elicit evidence that students have the ability to integrate knowledge and skills across multiple assessment targets and are ready to meet the challenges of college and careers.
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Items and tasks must be constructed at various levels of cognitive rigor. Smarter Balanced has defined four levels of depth of knowledge.
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The first level focuses on recall and reproduction of facts and other types of information.
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The second level focuses on basic skills and concepts that require cognitive processes that extend beyond the recall of information.
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The third level focuses on strategic thinking and reasoning.
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The fourth and final level requires extended thinking that includes complex reasoning, planning, development, and cognition that occurs over an extended period of time.
Let’s take a look at a sample item for each of the four levels of depth of knowledge.
This is a grade 8 item that is coded to depth of knowledge level one. This item requires students to recall the rules for exponents to evaluate each expression and select the expression or expressions with a value between zero and one.
This is a grade 8 item that is coded to depth of knowledge level two. This item requires students to use the formula for the volume of a cylinder, as well as a basic understanding of rate, to calculate the number of minutes Jane will take to fill the water tank.
This is a grade 8 item that is coded to depth of knowledge level three. This item requires students to perform calculations in order to respond to each part. In addition, students are required to provide an explanation for each answer.
This is a description for a grade 8 performance task that is coded to depth of knowledge level four. This item requires students to use concepts of geometry, numbers and operations, and statistics to determine the best solution to a problem where all constraints cannot be satisfied at the same time. Additionally, the student must provide justifications to support reasoning. Students are expected to engage with the task for an extended period of time, up to 120 minutes.
The depth of knowledge levels defined by Smarter Balanced are informed by the Cognitive Rigor matrix. The Cognitive Rigor matrix combines two common taxonomies that categorize levels of cognition and shows how the Smarter Balanced depth of knowledge categories relate to these taxonomies. Here, the concept of remembering information only relates to the first depth of knowledge level. In contrast, evaluation does not relate to either of the first two levels of depth of knowledge. This table, which can be found in the Smarter Balanced Content Specifications for Mathematics, is a useful aid for guiding the development of items at different depth of knowledge levels.
Now let’s examine the content specifications.
The content specifications define how the Consortium intends to assess the Common Core State Standards for Mathematics. The content specifications provide a translation between the grade level Common Core State Standards and a content framework that was used to establish item specifications.
The Mathematics Content Specifications describe the four claims and provide a set of assessment targets for each claim.
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The first claim is Concepts and Procedures, which requires students to explain and apply mathematical concepts to interpret and carry out mathematical procedures with precision and fluency.
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The second claim focuses on Problem Solving and requires students to solve complex mathematical problems using knowledge and problem solving strategies.
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Claim 3 addresses communicating reasoning and requires students to clearly and precisely construct viable arguments to support their reasoning and to critique the reasoning of others.
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And Claim 4 focuses on modeling and data analysis and requires students to analyze complex, real-world scenarios and construct and use mathematical models to interpret and solve problems.
Let’s take a closer look at an example of an assessment target.
The number of Claim 1 assessment targets varies across grade levels. The assessment targets for each grade level are presented in detail in the Content Specifications and are explored in greater detail in the Grade Level Considerations training modules. For now, note that careful thought went into examining the progression of mathematical knowledge and skills as students progress from early elementary grades through high school and as students develop college and career readiness. This progression informed the development of each grade level assessment target. Also note that for high school, assessment targets are established for grade 11 only and reflect the skills and knowledge students are expected to demonstrate in order to be college and career ready.
To help use assessment targets to inform the development and review of items and tasks, let’s take a closer look at the structure of an assessment target.
This is a Claim 1 assessment target for grade four. As described earlier, the standards are grouped into clusters. The assessment targets are organized using the same cluster headings defined in the Common Core State Standards.
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In this example, Target A falls under the domain, Operations and Algebraic Thinking. Depending on the grade level, there may be one or more assessment targets for each domain or conceptual category
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Each assessment target is labeled with a letter. This example shows Target A.
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Recognizing the design principles of focus and coherence in the standards as a whole, not all content is emphasized equally in the Standards for Mathematical Content. The standards communicate this emphasis through the use of domain names that vary across the grades and through the progressions that point to the major work of each grade. The Mathematics Content Specifications identify assessment targets that point to the “major” work of each grade with an “m” and other assessment targets as “s” for “supporting” and “a” for “additional.” Evidence for Claim 1 will strongly focus on the “major” clusters and take into account ways in which the standards tie “supporting” clusters to the major work of each grade.
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Next, a descriptive title of the assessment target is provided. In this example, the title is “Use the four operations with whole numbers to solve problems.”
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The depth of knowledge level to which items and tasks that measure the assessment target are to be written is indicated. In this case, items and tasks that measure 4th grade Target A should tap level one and level two depths of knowledge.
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Finally, a full description of the assessment target is provided. The description provides details on the mathematical concepts, skills, and knowledge required by the assessment target and describes the types of mathematical content and problems students are expected to work with and complete successfully.
Let’s take a few moments to look briefly at the assessment targets for Claim 1 for each grade level. A full description of each target can be found in the Content Specifications.
There are eleven assessment targets specified for Grade 3.
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The first four targets are classified under the Operations and Algebraic Thinking domain.
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A single target falls within Number and Operations – Base 10.
Similarly, one target falls under the Numbers and Operations – Fractions domain.
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The four targets are within the Measurement and Data domain.
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And one target falls within Geometry.
For grade 4, there are twelve assessment targets.
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The first five targets are classified under the Operations and Algebraic Thinking domain.
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Three targets fall within Number and Operations – Fractions.
Under the Measurement and Data domain, there are three targets.
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And a single target falls within the Geometry domain.
For grade 5, there are eleven assessment targets.
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The first two targets are classified under the Operations and Algebraic Thinking domain.
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Two targets fall within Number and Operations – Base 10.
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Two targets also fall within Number and Operations – Fractions.
There are three targets within the Measurement and Data domain.
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And two targets fall within Geometry.
For grade 6, there are ten assessment targets.
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The first target is classified under the Ratios and Proportional Relationships domain.
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Three targets fall within the Number System domain.
The Expressions and Equations domain contains three targets.
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A single factor is specified for the Geometry domain.
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And two targets are specified for Statistics and Probability.
For grade 7, there are nine assessment targets.
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The first target is classified under the Ratios and Proportional Relationships domain.
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One target falls within the Number System domain.
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Two targets are specified for Expressions and Equations.
The Geometry domain contains two targets.
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And there are three targets specified for Statistics and Probability.
For grade 8, there are ten assessment targets.
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The first target is classified under the Number Systems domain.
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Three targets fall within Expressions and Equations.
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Two targets are specified for Functions.
For Geometry, three targets are specified.
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And for Statistics and Probability there is one target.
Finally, for grade 11, there are sixteen assessment targets.
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The first three targets are classified under the Number and Quantity conceptual category.
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Seven targets fall within Algebra.
For the Functions conceptual category, there are four targets specified.
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There is one target specified for Geometry.
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Finally, there is one target specified for Statistics and Probability.
Participants should spend time reviewing the assessment targets for the grade levels for which they will be developing or reviewing items and tasks.
Now that the assessment targets for Claim 1 have been briefly presented for each grade level, let’s shift our focus to Claims 2 through 4. Unlike the Claim 1 assessment targets, that are drawn from the grade-level Standards for Mathematical Content, the assessment targets for Claims 2, 3 and 4 are drawn from the Standards for Mathematical Practice that are identical across grade levels.
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A general set of assessment targets is provided in the Content Specifications and can be used as guidance for the development of item and task specifications for all grade levels. Let’s look at the assessment targets for Claims 2, 3, and 4.
Claim 2 focuses on Problem Solving. The purpose of this claim is to elicit evidence that students can solve a range of complex well-posed problems in pure and applied mathematics and can make productive use of knowledge and problem solving strategies. Items and tasks written to assessment targets for this claim will ask students to:
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Apply mathematics to solve well-posed problems arising in everyday life;
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select and use tools strategically;
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Interpret results in the context of a situation;
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And identify important quantities in practical situations and to map their relationships.
Items and tasks written for Claim 2 will provide evidence for several of the Claim 2 assessment targets. Each target should not lead to a separate task: it is in using content from different areas, including work studied in earlier grades, that students demonstrate their problem solving proficiency.”
Claim 3 focuses on Communicating Reasoning. The purpose of this claim is to elicit evidence that students can clearly and precisely construct viable arguments to support their own reasoning and to critique the reasoning of others. Items and tasks written for this claim will ask students to explain his or her reasoning, justify a conjecture, and assess the validity of a claim.
The Claim 3 targets require students to:
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Test propositions or conjectures;
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Construct chains of reasoning that justify or refute propositions or conjectures;
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State logical assumptions that are made;
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Use techniques of breaking arguments into cases;
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Distinguish correct logic and flawed reasoning and explain what it is
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Base arguments on concrete referents
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And determine conditions under which an argument does and does not apply.
Items and tasks written for Claim 3 will provide evidence for several of the Claim 3 assessment targets. Each target should not lead to a separate task. Tasks generating evidence for Claim 3 in a given grade will draw upon knowledge and skills articulated in the standards in that same grade, with strong emphasis on the major work of the grade.”
Claim 4 focuses on modeling and data analysis. Claim four requires extended response items and performance tasks that elicit evidence that students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. Items and tasks written for this claim will ask students to investigate problems that have more than one solution pathway, summarize the results within the context of the problem, and evaluate the solution within the context of the problem.
The assessment targets associated with Claim 4 require students to:
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Apply mathematics to solve problems arising in everyday life;
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Construct chains of reasoning to justify mathematical models used, interpretations made, and solutions proposed for complex problems;
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State logical assumptions that are made;
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Interpret results in the context of a situation;
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Analyze the adequacy of and make improvements to an existing model or develop a mathematical model of a real phenomenon;
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Identify important quantities in a practical situation and map their relationships;
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And identify, analyze, and synthesize relevant resources to pose or solve problems.
Items and tasks written for Claim 4 will provide evidence for several of the Claim 4 assessment targets. Each target should not lead to a separate task. Tasks generating evidence for Claim 4 in a given grade will draw upon knowledge and skills articulated in the progression of standards up to that grade, with strong emphasis on the “major” work of the grades.
Now, let’s shift our focus from the claims and assessment targets specified in the Content Specifications to the additional information presented in the Item Specifications.
The Smarter Balanced Assessment Consortium Mathematics Item and Task Specifications are a bridge between the mathematics content specifications and the summative assessments.
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The item specifications further define each claim and assessment target in the content specifications by identifying evidence statements for each target.
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Each evidence statement has one or more associated task models. The task model describes how an item that fits the evidence statement, and thus the assessment target, are to be constructed.
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Sample items are provided as additional guidance for item and task writers. The sample items illustrate how the task models can be operationalized in an assessment item.
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A glossary of terms has also been provided.
Let’s explore the item and task specifications in more detail.
The Item and Task Specifications contain a large number of item specification tables. Each table focuses on a specific claim and assessment target or targets, and presents information that is used to guide the writing of items and tasks that collect evidence about that claim and assessment target. Let’s take a moment to examine several key parts of an item specification table.
{+} The top portion of a table indicates the grade, claim, and assessment target addressed by the table. In this example, the grade is three, and the table addresses claim one, assessment target A.
{+} The first row restates the claim.
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The second row states the content domain.
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The third row indicates the assessment target
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And the fourth row indicates which standards are associated with the assessment target. All of this information is based on information provided in the Content Specifications. {+}
The fifth row indicates the depth of knowledge level at which the assessment target can be assessed. In this example, level one and level two are both assessable.
{+} The sixth row provides detailed information about the type of evidence required to support the claim about the assessment target. This information is useful for informing the contents of the prompt developed to elicit evidence from students.
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The seventh row indicates the item types that can be used to elicit evidence from students. In this example, selected response, constructed response, and technology enhanced items are deemed appropriate.
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Next, task models describe the characteristics of items and tasks that may be used to collect evidence. There will be more about task models in a few moments.
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The ninth row provides information about the types of stimulus materials that should accompany the item. In this example, arrays or equal group models are expected to be presented with the item prompt.
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The tenth row details vocabulary that is associated with the assessment target and which can be used as part of the item’s content. This section provides guidance for making decisions about whether or not a term might introduce bias. Any term listed in this section is deemed appropriate for use within the context of the items or tasks developed for the claim and assessment target.
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The eleventh row indicates which tools, if any, can be used while working on the item or to produce responses for the item. This information is intended to support decisions about potential accommodations and about resources that should be available to students as they work on this item.
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Next, information about any specific attributes of the item are provided. In this example, it is stated that numbers used in problems must be less than or equal to 100.
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The thirteenth row identifies any non-targeted knowledge or skills that may be required to respond to items based on the task models.
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Any accessibility concerns that are specific to any of the task models is then documented. This information is used to help inform decisions about accommodations for the item or tasks built from the task model.
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Finally, if a sample item exists, the next section will provide a link to it. Sample items provide models for what an item or task might look like.
With that overview of an item specification table, let’s take a closer look at an evidence statement.
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This is an evidence statement for grade 5 , Claim 1, Assessment Target C,
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which focuses on understanding the place value system.
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The evidence statement identifies one piece of evidence required for this claim and assessment target. Specifically, students must be able to explain the pattern in the number of zeros in a product when multiplying a number by a power of ten.
Evidence statements are used to inform the types and content of responses collected from students through an item or task, and provide a foundation for developing Task Models.
A task model describes key features and characteristics of items or tasks that are designed to elicit a specific type of evidence for a given assessment target and claim. This is a task model for the evidence statement just discussed.
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The task model begins by specifying a constructed response item.
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And indicates that the items should require students to apply knowledge and skills at a level one or two depth of knowledge.
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The task model then describes the features of prompts. In this example, the prompt is expected to ask students to explain patterns in the number of zeros of a product when multiplying a number by a power of ten.
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Next, information about stimuli that are to accompany the prompt is provided. In this example, the stimuli are expected to contain multiplication problems involving powers of ten.
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For many evidence statements, more than one task model might be specified. In this example, a second prompt and stimuli are described, which in this case focuses on quotients rather than products.
One expectation of some items written for the Consortium is that the item will assess multiple claims and assessment targets. This is particularly true for items written for Claims 2, 3, and 4. This is also true for certain item types, such as extended-response items and performance tasks.
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As an example, this item table is for a grade 7 constructed response item.
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For this item, the Primary Claim addressed by the item is Claim 2, which focuses on problem solving.
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This item also provides information about Claim 1, which focuses on concepts and procedures.
When items or tasks address more than one claim or more than one target within a claim, the claims and assessment targets should be identified in order of importance with regards to the item.
As this module has explored, the Common Core State Standards provide a foundation for defining the claims and assessment targets that are the focus of the Smarter Balanced Assessment System.
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The content specifications were developed by Smarter Balanced to provide a formal definition of the claims and assessment targets and provide a foundation for item specifications.
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In addition, the content specifications indicate the depth of knowledge that students are expected to demonstrate for a given assessment target.
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The item and task specifications were also developed by Smarter Balanced to define the evidence to be elicited from students to support claims about assessment targets and present task models that inform the development of items that are used to collect that evidence.
Collectively, this information is used to guide the development of items and tasks that form the Smarter Balanced Assessment System. To support the development of high quality items and tasks, item writers need to be familiar with the content and item specifications and are expected to reference them regularly when developing items and tasks.
