Test Design and 
Test Framework

Field 314: Elementary Mathematics Specialist

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The test design below describes general test information. The framework that follows is a detailed outline that explains the knowledge and skills that this test measures.

Test Design

Format	Computer-based test (CBT)
Number of Questions	80 multiple-choice questions
Time*	3 hours
Passing Score	240

*Does not include 15-minute CBT tutorial

Test Framework

Subarea 1—Number Concepts and Operations

Objective 0001—Apply knowledge of number sense, number systems, and the properties of the real number system.

For example:

• Apply knowledge of prenumber concepts (e.g., sorting, ordering, one-to- one correspondence, cardinality, composition and decomposition), subitizing, place value, and manipulations of numbers.
• Apply knowledge of how to represent, compare, and order numbers using a variety of models (e.g., number lines, base-ten blocks, diagrams).
• Apply knowledge of representations of equivalent rational numbers (e.g., fractions, decimals, percents) and converting between them in mathematical and real-world situations.
• Analyze the characteristics of numbers (e.g., odd and even, positive or negative) and the sets of whole numbers, integers, rational numbers (e.g., fractions, decimals, percents, exponents), irrational numbers, and real numbers.
• Apply knowledge of ways to support student learning with the use of academic language and vocabulary related to number sense, the structure of number systems, and the properties of the real number system.
• Recognize common student misconceptions and partial understandings related to number sense, the structure of number systems, and the properties of the real number system and identify and apply appropriate strategies to develop student understanding.

Objective 0002—Analyze number operations and computational algorithms.

For example:

• Analyze and apply basic number concepts (e.g., skip counting, composition/decomposition, factors/multiples, prime/composite numbers, least common multiple/greatest common factor) to make sense of computational algorithms.
• Analyze and apply operations and evaluate the appropriateness of the algorithms based on their advantages and limitations.
• Apply knowledge of relationships between operations (e.g., repeated addition and multiplication, inverse operations) to analyze and understand student thinking.
• Analyze and apply estimation techniques and mental math strategies to real-world problems involving integers, fractions, decimals, and percents.
• Analyze and apply properties of operations (i.e., commutative, associative, distributive, and identity) and the order of operations to justify procedures, solve problems, and evaluate basic algebraic expressions and equations.
• Analyze multiple algorithms, strategies, and representations (e.g., rectangular arrays, partitioning, decomposing) of basic operations with whole numbers, fractions, and decimals.
• Solve a variety of mathematical and real-world problems using whole numbers, integers, fractions, decimals, percents, roots, powers, and rational exponents.
• Apply knowledge of ways to support student learning with the use of academic language and vocabulary related to number operations and computational algorithms.
• Recognize common student misconceptions and partial understandings related to number operations and computational algorithms and identify and apply appropriate strategies to develop student understanding.

 

Subarea 2—Algebra and Functions

Objective 0003—Apply knowledge of patterns, algebraic and proportional reasoning, expressions, and equations.

For example:

• Analyze and extend a variety of patterns (e.g., numbers, figures, expressions) and use a variety of number patterns to explore number properties.
• Analyze and make connections among various representations of sequences (e.g., verbal, written, numeric, graphic, symbolic).
• Represent and use proportional reasoning (e.g., ratios and proportions, percentages) to solve real-world and mathematical problems.
• Manipulate and simplify algebraic expressions (e.g., order of operations, factoring, distributive property, combining like terms) and solve equations and inequalities in both mathematical and real-world problems.
• Evaluate and justify the rationale for the manipulation of algebraic expressions, equations, and inequalities.
• Analyze mathematical and real-world problems and translate them into algebraic expressions and equations.
• Apply knowledge of ways to support student learning related to patterns, algebraic and proportional reasoning, expressions, and equations with the use of academic language and vocabulary.
• Recognize common student misconceptions and partial understandings related to patterns, algebraic and proportional reasoning, expressions, and equations and identify and apply appropriate strategies to develop student understanding.

Objective 0004—Apply knowledge and concepts of linear functions to model and solve problems.

For example:

• Demonstrate knowledge of the attributes of functions and relations (e.g., domain, one-to-one, inverse) and multiple representations (e.g., graphic, verbal, algebraic) of them.
• Identify and analyze the relationships between linear functions, proportions, and average rate of change.
• Model and solve mathematical and real-world problems involving linear functions using a variety of representations and strategies (e.g., tabular, graphic, algebraic).
• Analyze and solve problems involving linear functions, equations, and inequalities using a variety of representations and strategies (e.g., algebraic, graphic, tabular, verbal).
• Apply knowledge of ways to support student learning related to modeling and problem-solving with linear functions with the use of academic language and vocabulary.
• Recognize common student misconceptions and partial understandings related to modeling and problem-solving with linear functions and identify and apply appropriate strategies to develop student understanding.

 

Subarea 3—Geometry and Measurement

Objective 0005—Apply knowledge and concepts of Euclidean and coordinate geometry.

For example:

• Analyze and apply knowledge of shapes and their attributes to construct, classify, and compare figures.
• Apply knowledge of ways to support student learning of geometrical concepts with the use of academic language and vocabulary.
• Analyze and explain the relationships between, and composition and decomposition of, two-dimensional and three-dimensional geometric figures.
• Analyze three-dimensional figures using two-dimensional representations (i.e., nets).
• Use concepts of geometry (e.g., congruence, symmetry, similarity, parallel and perpendicular lines) to solve mathematical and real-world problems.
• Construct lines, geometric figures, and polygons in the coordinate plane.
• Recognize common student misconceptions and partial understandings related to concepts of Euclidean and coordinate geometry and identify and apply appropriate strategies to develop student understanding.

Objective 0006—Apply knowledge and concepts of measurement.

For example:

• Apply knowledge of how to use customary systems (e.g., in., ft, yd) and metric systems (e.g., cm, m, km) appropriately and convert within them.
• Apply knowledge of ways to support student learning related to concepts of measurement with the use of academic language and vocabulary.
• Use dimensional analysis to represent and solve problems in a variety of situations.
• Analyze and solve a variety of measurement problems (e.g., length, perimeter, circumference, angles, area and surface area, volume, temperature, elapsed time).
• Apply and analyze precision, estimation, and rounding in measurements and computed quantities.
• Recognize common student misconceptions and partial understandings related to concepts of measurement and identify and apply appropriate strategies to develop student understanding.

 

Subarea 4—Data Analysis and Probability

Objective 0007—Analyze and interpret data.

For example:

• Apply knowledge of how to compare, organize, display, and analyze data in a variety of representations (e.g., bar graph, frequency distribution, box plot, histogram).
• Apply knowledge of ways to support student learning related to the analysis and interpretation of data with the use of academic language and vocabulary.
• Apply concepts of central tendency (e.g., mean, median, mode) and variability (e.g., range, outliers) in data sets and data distributions.
• Apply knowledge of how to describe and summarize data for the purpose of making decisions, predicting, and solving real-world problems.
• Analyze experimental designs (including exploration of statistical questions), interpret results, and draw inferences from observations and experiments that investigate real-world problems.
• Recognize common student misconceptions and partial understandings related to the analysis and interpretation of data and identify and apply appropriate strategies to develop student understanding.

Objective 0008—Apply knowledge and concepts of probability.

For example:

• Identify the appropriate sample space in problems involving probability.
• Demonstrate knowledge of how concepts of probability are used to solve problems involving simple events.
• Calculate probabilities and solve problems involving the counting principle (e.g., counting techniques, combinations).
• Represent and solve problems using multiple representations (e.g., tree diagrams, Venn diagrams) of real-world situations.
• Apply knowledge of ways to support student learning related to concepts of probability with the use of academic language and vocabulary.
• Recognize common student misconceptions and partial understandings related to concepts of probability and identify and apply appropriate strategies to develop student understanding.

 

Subarea 5—Professional Knowledge and Instructional Leadership

Objective 0009—Demonstrate knowledge of mathematics instruction and assessment.

For example:

• Demonstrate knowledge of ways to promote equity and access for all students in mathematics instruction.
• Demonstrate knowledge of how tools (e.g., manipulatives, technology) can be used to enhance student understanding.
• Analyze and use results from various types of assessments (e.g., diagnostic, formative, summative) to plan, inform, and adjust instruction.
• Apply knowledge of levels of questioning to assess students' mathematical understanding to advance their mathematical learning.
• Apply knowledge of ways to support student learning with the use of academic and pedagogic language and vocabulary.
• Apply knowledge of sequences of instruction and problem-solving tasks that develop students' content knowledge, reasoning skills, conceptual understanding, and computational fluency and precision.
• Integrate knowledge of the vertical alignment of mathematical topics and concepts across grade levels to plan instruction based on state standards.

Objective 0010—Demonstrate knowledge of instructional leadership in mathematics.

For example:

• Identify and apply ways to establish a culture of collaboration regarding the use of data to plan, evaluate, and improve mathematics instruction and to promote positive changes in the school mathematics program.
• Identify and apply ways to promote and support instructional programs based on research-supported best practices regarding mathematics curriculum, instruction, technology, and assessment.
• Identify and apply appropriate and effective methods for communicating professionally with educational stakeholders about students, curriculum, instruction, use of technology, and assessment.
• Demonstrate knowledge of ways of using professional resources (e.g., organizations, journals, discussion groups) to stay current regarding critical issues related to mathematical teaching and learning; enhancing one's own professional knowledge and skills; and engaging in reflective practices (e.g., evaluating and adjusting one's own performance in a variety of instructional contexts).
• Demonstrate knowledge of educational structures and policies that affect students' equitable access to quality mathematics instruction and encourage the use of practices with proven effectiveness in promoting academic success for students with diverse characteristics and needs.
• Analyze and apply knowledge of strategies for collaborating effectively with families/caregivers and community members to support students' mathematical development (e.g., partnering with families/caregivers and community members in promoting students' lifelong appreciation of mathematics; communicating findings of current research in mathematical development to various stakeholders, including families/caregivers, local libraries, businesses, and policymakers) and for supporting positive family/caregiver�student interactions related to mathematics.
• Analyze and apply knowledge of ways to collaborate effectively with colleagues to promote professional development and to meet the mathematics needs of all students (e.g., coaching, conducting professional study groups for teachers, providing constructive feedback on colleagues' practices related to mathematics instruction).
• Analyze and apply knowledge of components and procedures related to effective, evidence-based, and Multi-Tiered System of Supports (MTSS) used in Illinois.
• Integrate knowledge of ways to use professional development (e.g., mentoring, coaching, peer-teaching, workshops) to facilitate appropriate research-supported, standards-based mathematics instruction and to promote the use of instructional methods supported by research.

 

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