Mathematics Content Specifications, Item Specifications, and Depth of Knowledge Training Module

Mathematics Content Specifications, Item Specifications, and Depth of

Knowledge Training Module

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

Foundation for Item and Task Development

Common Core State Standards

Smarter Balanced Content Specifications

Smarter Balanced Item and Task Specifications

Items and Performance Tasks

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

Structure of theCommon Core State Standards for Mathematics

STANDARD

CLUSTER

DOMAIN Number and Operations in Base Ten 3.NBTUse 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..

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

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.

Level 1 ExampleGrade 8

Select all of the expressions that have a value between 0 and 1. 87 ∙ 8–12

74

7–3

13

2

∙ 13

9

(–5)6

(–5)10


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.


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.


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.

Cognitive Rigor MatrixThis 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.

Smarter Balanced Assessment Consortium Mathematics Content Specifications

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

Claim 1Concepts and Procedures

Students can explain and apply mathematical concepts and interpret and carry out mathematical procedures with precision and fluency.

Grade LevelNumber of

Assessment Targets3 11

4 12

5 11

6 10

7 9

8 10

11 16

Claim 1 Concepts and Procedures

Grade 4Operations 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).

Claim 1 Assessment TargetsGrade 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 10E. Use place value understanding and properties of arithmetic to

perform multi-digit arithmetic.


Numbers and Operations – FractionsF. 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.


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 – FractionsF. 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.


Measurement and DataI. 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.

GeometryL. Draw and identify lines and angles, and classify shapes by

properties of their lines and angles.


Operations and Algebraic Thinking A. Write and interpret numerical expressions. B. Analyze patterns and relationships.

Number and Operations – Base 10C. Understand the place value system. D. Perform operations with multi-digit whole numbers and with

decimals to hundredths.

Number and Operations – FractionsE. 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.


Measurement and DataG. 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.

GeometryJ. Graph points on the coordinate plane to solve real-world and

mathematical problems. K. Classify two-dimensional figures into categories based on their

properties.


Ratios and Proportional RelationshipsA. Understand ratio concepts and use ratio reasoning to solve

problems.

The Number SystemB. 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.


Expressions and EquationsE. 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.

GeometryH. Solve real-world and mathematical problems involving area,

surface area, and volume.

Statistics and ProbabilityI. Develop understanding of statistical variability. J. Summarize and describe distributions.


Ratios and Proportional RelationshipsA. Analyze proportional relationships and use them to solve real-

world and mathematical problems.

The Number SystemB. Apply and extend previous understandings of operations with

fractions to add, subtract, multiply, and divide rational numbers.

Expressions and EquationsC. Use properties of operations to generate equivalent expressions.D. Solve real-life and mathematical problems using numerical and

algebraic expressions and equations.


GeometryE. 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 ProbabilityG. 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.


The Number SystemA. Know that there are numbers that are not rational, and

approximate them by rational numbers.

Expressions and EquationsB. 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.

FunctionsE. Define, evaluate, and compare functions. F. Use functions to model relationships between quantities.


GeometryG. 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 ProbabilityJ. Investigate patterns of association in bivariate data.


Number and QuantityA. 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.

AlgebraD. 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.


FunctionsK. 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.

GeometryO. Prove geometric theorems.

Statistics and ProbabilityP. Summarize, represent and interpret data on a single count or

measurement variable.

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.

Assessment TargetsClaim 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.

Assessment TargetsClaim 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.

Assessment TargetsClaim 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.

Structure of the Smarter Balanced Assessment Consortium Mathematics

Item and Task Specifications

• Evidence Statements

• Task Models

• Sample Items

• Mathematics Glossary

Structure of Item Specification Table

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.

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.

Alignment to Claims and Assessment Targets

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

Documents

Mathematics Content Specifications, Item Specifications, and Depth of Knowledge Training Module