Study Guide
Overview and Test Objectives
Field 133: Mathematics (5–9)
Test Overview
| Format | Computer-based test (CBT) |
|---|---|
| Number of Questions | 100 multiple-choice questions |
| Time | 2 hours 30 minutes* |
| Passing Score | 220 |
*Does not include 15-minute CBT tutorial
Test Objectives
| Subarea | Range of Objectives | Approximate Percentage of Questions on Test | |
|---|---|---|---|
| 1 | Pedagogical Knowledge and Practices for Teaching Mathematics | 001–004 | 18% |
| 2 | Students as Learners of Mathematics | 005–007 | 13% |
| 3 | Understanding of Social Contexts of Mathematics Teaching and Learning | 008–010 | 13% |
| 4 | Know Relevant Mathematical Content | 011–018 | 38% |
| 5 | Mathematics Practices, Dispositions, Curriculum | 019–022 | 18% |
Hover over each subarea for details of subtest content or see table above.
Sub area 1 18%, Sub area 2 13%, Sub area 3 13%, Sub area 4 38%, and Sub area 5 18%.
Subarea 1—Pedagogical Knowledge and Practices for Teaching Mathematics
Objective 001—Promote Equitable Teaching
Includes:
- Facilitate a range of tasks through equity-based pedagogy including consideration of students' individual needs, cultural experiences, and interests, as well as prior mathematical knowledge.
- Develop a classroom community in which students present ideas, challenge one another's ideas respectfully, construct meaning together, value and celebrate varied mathematical strengths, and use mathematics to address problems and issues in their school, homes, and communities.
- Ensure all student approaches, responses, representations, experiences, and voices are valued in mathematical inquiries, discourse, and problem-solving.
- Facilitate multiple opportunities for all students to formulate, represent, analyze, and interpret mathematical models using a variety of tools including technology.
- Provide all students access to the ways of doing mathematics (e.g., inquiry, technology, mathematical language including symbols and notation).
- Engage all students in challenging mathematics content, building from their own funds of knowledge as they use multiple representations and models of their choice.
- Use students' developing understandings as found in various student representations (e.g., visualizations, vocalizations, models, symbols, notation) to appropriately plan next steps for instruction.
Objective 002—Plan for Effective Instruction
Includes:
- Establish appropriate and rigorous learning goals for students that build on student understandings and are situated within learning progressions, based on research about student learning, mathematics standards and practices, and the approach to learning mathematics.
- Attend to the development of both conceptual and procedural understanding to choose tasks and design instruction.
- Plan and implement rich tasks, including the appropriate instructional strategies that provide opportunities and access for all students to actively engage in the mathematical learning.
- Anticipate an array of students' responses to tasks and craft questions, and prepare follow-up replies to explore student thinking in a way that relates the mathematical concepts and procedures.
- Select mathematics-specific tools and technology to develop students' conceptual understanding of mathematics.
- Plan ways to use evidence of student thinking to assess progress toward mathematical understanding and possible instructional adjustments.
- Draw on current research to develop mathematics instruction and assessment.
- Consider how to motivate and engage all students in learning mathematics.
Objective 003—Implement Effective Instruction
Includes:
- Use established learning goals to guide instructional decisions.
- Engage students in solving and discussing tasks that promote mathematical reasoning and problem-solving and allow multiple entry points and varied solution strategies.
- Engage students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and to use as a tool for problem-solving.
- Facilitate discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student strategies and arguments.
- Pose purposeful questions to assess and advance students' reasoning and sense-making about important mathematical ideas and relationships.
- Use strategies that ground procedural fluency in conceptual understanding so that students, over time, become skillful in using procedures flexibly and efficiently as they solve contextual and mathematical problems.
- Provide students with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships.
- Use evidence of student thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning.
- Analyze students' thinking that leads to an incorrect answer, identify the mathematical understanding that students have and may not yet have, and reply in a way that helps students develop their own understanding.
- Implement the use of appropriate mathematical tools (e.g., technology and manipulatives) to develop students' conceptual understanding.
- Enact effective mathematics instruction that supports the learning of each and every student, including appreciating and accepting student reasoning that may be atypical or different from their own.
- Reflect upon instruction to inform planning for future teaching.
Objective 004—Enhance Teaching Through Collaboration with Colleagues, Families, and Community
Includes:
- Make professional ideas and decisions visible and subject to review by colleagues to develop a deeper shared understanding of students' learning and to support their development as a teacher.
- Plan strategies to grow professionally and promote the mathematical success of students in collaboration with colleagues.
- Understand, interact, and build intentional relationships with families, caregivers, and community members to learn how their perspectives, priorities, and cultures can provide information regarding the mathematical learning needs of students.
- Utilize mathematics knowledge and experiences related to students' identity, culture, community, daily life, and history as resources for interactions with students.
- Provide constructive feedback to students' families and caregivers focused on strengths and areas of growth concerning students' mathematical performance.
- Develop shared strategies with families and caregivers for promoting mathematical success of students.
- Possess an awareness of and commitment to employing multiple strategies to get to know students' families, caregivers, and communities in order to better serve students.
Subarea 2—Students as Learners of Mathematics
Objective 005—Anticipate and Attend to Students' Thinking About Mathematics Content
Includes:
- Understand learning progressions within the mathematical content domains, which includes knowledge of the mathematics that comes before and after a given mathematics topic.
- Elicit and analyze students' thinking to understand where students lie on the learning progression.
- Utilize understanding of students' thinking to plan for and execute instructional moves to advance students' learning.
- Plan frequent opportunities for students to be metacognitive about their own learning and understandings.
- Recognize the importance of eliciting and understanding student experiences and identity in shaping their mathematical thinking.
Objective 006—Promote Students' Engagement in Mathematical Practices
Includes:
- Anticipate that students will present various approaches to problems and support students in analyzing, comparing, justifying, and proving their solutions.
- Create environments in which strategies are respectfully discussed, critiqued, and compared.
- Identify how contexts, culture, conditions, and language can be used to create meaningful and authentic tasks that relate to students' interests, backgrounds, prior knowledge, and experience, leading to increased engagement and motivation in mathematics.
- Present tasks that require high cognitive demand; have multiple solution strategies, entry points, and representations; require communication of thinking and reasoning; and ensure students engage in rigorous mathematics.
- Reflect on teaching moves that support or inhibit the engagement of students with the mathematical practices.
- Pose questions to students that help them analyze problem situations, select strategies, and reason quantitatively.
Objective 007—Cultivate Positive Mathematical Identities of Students
Includes:
- Understand that the teacher has a central role in student learning of mathematics and that teachers should view their roles as supporting the development of students' mathematical identities through their interactions with students and instructional decisions.
- Plan and implement mathematics instruction that draws on all students' mathematical strengths and positive mathematical identities that will allow them to be successful with the mathematics they are learning, which in turn continues to develop positive mathematical identities.
- Analyze students' task selection and implementation, reflecting on ways this shapes students' mathematical identities, and consider how the experience of doing the task supports developing a positive mathematical identity for each student.
- Create classroom environments and orchestrate classroom discussions that enable respectful communication about mathematical ideas that support the development of positive student identities.
- Work to counter negative beliefs including stereotypes about who is good at mathematics and build positive beliefs within and among students.
- Exhibit an asset-based perspective rather than a deficit-based view in interactions with students, realizing that mathematical errors are opportunities for learning and that all students bring their own unique mathematical strengths to the learning environment.
- Acknowledge the diversity of all individual and group identities, particularly those of students whose learning experiences and needs are different from their own, including both individual differences (e.g., personality, interests, learning modalities, life experiences) and group differences (e.g., race, ethnicity, ability, gender identity and fluidity, gender expression, sexual orientation, nationality, language, religion, political affiliation, socio-economic background) and use these in designing instruction to validate and build productive identities.
- Take conscious and intentional actions to build students' agency as mathematical learners, based on strong beliefs that each and every student can learn mathematics with understanding.
Subarea 3—Understanding of Social Contexts of Mathematics Teaching and Learning
Objective 008—Provide Access and Opportunity
Includes:
- Understand that access means ensuring all students, including those that have been historically marginalized, have qualified teachers, opportunities for placement into higher-level courses, high-quality curriculum, and opportunities to approach problems.
- Understand how denial of access and advancement perpetuates and produces inequities in the learning of mathematics for all students, including traditionally underrepresented and/or underserved students.
- Demonstrate knowledge of and advocate for equitable practices for identifying students for advancement, which include analysis of multiple indicators ensuring advancement is open to a wide range of students.
- Understand the negative impacts of tracking students into qualitatively different or dead-end course pathways.
- Understand and employ accommodations (formal and informal) available for students with exceptionalities (e.g., students with disabilities, English learners, advanced learners, students under duress) to promote their access and advancement in mathematics.
Objective 009—Understand Power and Privilege in the History of Mathematics Education
Includes:
- Understand current and historical mathematical educational practices that contribute to inequitable student opportunities and outcomes such as, but not limited to, social barriers (e.g., finances, teaching philosophy, demographic trends, culture), structural barriers (e.g., classroom size, schools, teachers, resources), and system policies (e.g., those related to placement and instruction, tracking, high-stakes test-taking).
- Demonstrate knowledge of national reform movements in mathematics education, including the strides and challenges in affording every student a quality mathematics education.
- Recognize ways to advocate for changes to policies and procedures that have negatively impacted mathematics learning, particularly for those students who have not historically experienced success in mathematics.
- Analyze mathematical curriculum and instruction to determine whether either is likely to contribute to inequitable mathematical outcomes and opportunities for students.
- Recognize implicit and explicit biases in teachers and others, including biases in the school/district culture, which work against equitable mathematics learning opportunities and supports for all learners, and work to counter these biases so that all students can learn challenging mathematics deeply and well.
Objective 010—Enact Ethical Practices for Advocacy
Includes:
- Develop and use language that is effective in advocating for all students and conveys high expectations for learning mathematics.
- Identify personal beliefs, classroom practices, and systemic structures that produce equitable and inequitable mathematical learning experiences and outcomes for students.
- Work with others to develop strategies and methods to ensure traditionally marginalized students experience success in mathematics.
- Use effective advocacy strategies that promote meaningful inclusion of all students in the learning of mathematics.
- Recognize teachers' responsibility to stand up to exclusion, prejudice, and injustice affecting students in the learning of meaningful and robust mathematics.
Subarea 4—Know Relevant Mathematical Content
Objective 011—Understand Essential Concepts in Number
Includes:
- Demonstrate number sense: flexible reasoning with and about whole numbers, integers, and rational numbers in a variety of situations and applications through opportunities such as composing and decomposing numbers and number talks.
- Describe the underlying structure of the real number system and the learning progression for the development of number across the grades from kindergarten through high school.
- Identify and apply a variety of strategies to compare and estimate rational and irrational numbers.
- Understand and be fluent in using operations and appropriate notation, including exponentiation, with rational numbers, and apply and justify multiple strategies for adding, subtracting, multiplying, and dividing rational numbers.
- Apply and connect concepts such as factor, prime, divisible, and multiple to particular numbers and sets of numbers.
- Reason about and prove basic theorems about real numbers (e.g., the product of two negative numbers is positive, √2 is irrational, the product of two odd numbers will be odd).
- Use technology to investigate certain numbers (e.g., value of pi, compare the relative size of two numerical expressions, evaluate limiting processes).
- Understand how complex numbers are related to the solutions of quadratic equations.
Objective 012—Understand Essential Concepts in Ratios and Proportional Relationships
Includes:
- Describe the learning progression for the development of proportional reasoning across the grades from kindergarten through high school.
- Identify situations in which ratios can be a tool to solve problems and apply a variety of strategies such as ratio tables, double number lines, and unit rates to solve problems involving ratios.
- Understand ratios as paired quantities that vary together in the same relationship, distinct from a single number, recognizing that a ratio a to b may be associated with a value a over b (if b is not 0) and describe the differences and similarities between ratios and fractions.
- Identify and use equivalent ratios, reason about the role of multiplication and addition in generating equivalent ratios, and recognize that the sum of equivalent ratios is another equivalent ratio.
- Associate a unit rate with a ratio, recognize that equivalent ratios have the same unit rate, and recognize this unit rate as the constant of proportionality (k) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
- Connect the constant of proportionality (k) to the slope of a line through the origin and to the equations of such a line ratio a to b arrow y equals open parens b over a close parens x, and describe how this perspective relates to the general concept of linearity.
- Reason about contextual situations, identifying those that involve proportional relationships, and use a variety of strategies for solving problems involving proportions, including finding a unit rate.
- Connect ratios to scale factors, both within and between scaled figures, and relate scale factors to similar shapes, including how scaling a figure affects the area and volume of the scaled figures.
- Describe percentage as a particular ratio of a quantity to 100, and apply this understanding to solve a variety of contextual problems.
Objective 013—Understand Essential Concepts in Algebra at a Comprehensive, Robust Level
Includes:
- Explain how algebra as the language of generalization is useful for describing patterns and relationships.
- Appreciate the value that new technologies can bring to developing algebraic understandings and strategically employ them in improving learning experiences in algebra for all students.
- Describe how algebraic concepts build from arithmetic and are connected to other content areas, such as geometry, statistics, and calculus.
- Describe the role of and be able to apply definitions, reasoning, and proof in algebra including identifying conditions under which theorems are valid, recognizing contradiction as a proof strategy, and using conjectures to investigate algebraic relationships.
- Use different technologies to enhance the learning of mathematics such as computer algebra systems to investigate algebraic structures and to check results, spreadsheets to produce and explore regularity in repeated reasoning, graphs to explore algebraic relationships, and interactive dynamic technologies to develop conceptual understanding of key algebraic topics.
- Interpret the structure of an algebraic expression in terms of a context and understand that structure can provide insight into a mathematical situation.
- Connect symbolic, graphical, tabular, and verbal representations of a problem or situation and explain any advantages and disadvantages of each representation for the given problem or situation.
- Use algebra as a tool to solve contextual problems including identifying variables, formulating an algebraic model, manipulating and analyzing the model, interpreting the results, and validating the conclusions.
- Explain and justify routine procedures involved in manipulating expressions and solving equations including the use of the properties related to multiplication, addition, and equality.
Objective 014—Understand Essential Concepts in Functions (e.g., Linear, Exponential, Polynomial, Absolute Value, Piecewise-Defined)
Includes:
- Recognize the value of function as the language and organizational structure in the analysis of mathematical relationships.
- Understand how algebra concepts are related to the ideas of function and that not all algebraic equations are functions.
- Represent functions, with and without technology, in a variety of ways including mapping diagrams, function notation, recursive definitions, tables, and graphs.
- Connect members of the same function family and identify distinguishing attributes (structure) common to all functions within that family.
- Compare function families and describe their similarities and differences. (e.g., linear functions are additive, exponential functions are multiplicative).
- Describe and reason about key features of the graphs of functions, using appropriate language (e.g., zeros, intercepts, rate of change, increasing/decreasing and maximum/minimum values); associate symbolic representations with these features and interpret them in both mathematical and real-world contexts.
- Model a wide variety of real situations using functions and understand the processes of making and changing assumptions, assigning variables, and finding solutions to contextual problems.
- Apply one or more function transformations to a representation (e.g., verbal, symbolic, graphical, tabular) of a function and explain the effects and results on other representations.
Objective 015—Understand Essential Concepts in Statistics and Probability
Includes:
- Describe the difference between the way conclusions are made in mathematics (which are deductive and deterministic) and in statistics (which are inductive and account for uncertainty) and appreciate that statistical reasoning is always grounded in a context in the presence of variability.
- Describe statistics as an investigative process of problem-solving and decision-making and explain how proficiency in statistical thinking matures as learners gain more knowledge and experience with variability (the developmental levels as outlined in the Guidelines for Assessment and Instruction in Statistics Education).
- Use real data with a context and purpose, hands-on activities, and active learning and technology to explore concepts and to manage and analyze data in developing understanding of statistical ideas.
- Understand and explain measures of center and spread.
- Identify the role of randomization and chance in determining the probability of events.
- Draw inferences about a population based on a sample in light of sampling variability using simulation-based techniques that lead to informal inference procedures.
- Evaluate reports based on data, reasoning critically and asking questions about the implementation of the statistical investigation process, and critique the ways in which numbers and graphical representations are used in the media, mathematical contexts, and everyday discussions.
- Connect the probability of an outcome to a long-run relative frequency of an outcome and investigate chance processes, developing, using, and evaluating probability models.
- Analyze and describe the association between variables and considering whether causation can be established.
- Design a basic observational study based on the statistical investigation process.
- Reflect upon data from implementation of classroom practices to enhance instruction.
Objective 016—Understand Essential Concepts in Geometry, Trigonometry, and Measurement
Includes:
- Recognize and value geometry as a lens to reason with ideas and to solve problems from real-world and mathematical domains.
- Connect geometry and measurement to other mathematical content areas such as to algebra when working in the coordinate plane, to ratios and proportional relationships when exploring scaling and scale drawings, to functions when exploring transformations, to number theory when exploring Pythagorean triples, and to calculus when finding the area under a curve.
- Recognize and describe the levels of geometric understanding (such as van Hiele) as it develops through a progression from investigations to more formal proof and reasoning.
- Explain, apply, and reason about the Pythagorean theorem (both the statement and the contrapositive); reason through proofs of the theorem based on similarity, area, or transformations; and describe the connections of the Pythagorean theorem to areas of mathematics such as coordinate geometry (e.g., distance formula, equations of circles) and trigonometry.
- Use a variety of tools including dynamic geometry software to investigate and understand variance and invariance of geometric objects and to make and test conjectures.
- Understand congruence and similarity in terms of transformations and solve problems involving congruence and similarity in multiple ways.
- Apply the transformation definition of congruence and similarity in terms of rigid motions to establish congruence and similarity criteria and use these criteria to prove theorems about triangles, quadrilaterals, and other geometric figures.
- Solve problems using transformations, coordinate geometry, or Euclidean geometry.
- Understand the role of definitions, postulates, and axioms and use them to describe relationships among angles, lines, and 2- and 3-dimensional shapes, including the parallel postulate and its connection to Euclidean geometries.
Objective 017—Understand Essential Concepts in Calculus at a Foundational, Basic Level
Includes:
- Recognize and value the importance of calculus to solve observed phenomenal problems involving change.
- Understand how ideas related to algebra, geometry, and functions are involved in understanding and applying calculus concepts.
- Connect the concepts of limit, derivative, and integration.
- Use dynamic interactive technology to develop conceptual understanding of key concepts such as derivative or mean value theorem, use graphing technology to analyze functions and their first and second derivatives, and develop notions of limit and understand how sequences and series behave.
- Apply foundational definitions, theorems, and concepts from calculus to solve mathematical and contextual problems, attending to the need to verify the hypotheses/conditions for using theorems.
- Interpret results in the context of a given situation when using calculus to solve a problem.
- Fluently use and interpret the notation involved in describing and working with limits, derivatives, and integrals.
- Connect graphical, algebraic, tabular, and verbal representations of a problem involving rates of change and approximations, understanding the advantages and limitations of each.
- Identify and explain foundational common underlying structures in concepts involving rate of change and approximations.
Objective 018—Understand Essential Concepts in Discrete Mathematics at a Foundational, Basic Level
Includes:
- Seek out opportunities to engage in solving problems leveraging strategies and tools from areas of discrete mathematics.
- Identify ways in which discrete mathematics can connect different mathematical domains.
- Use a variety of techniques to count and arrange sets of objects (combinatorics), including making the connection of counting to Pascal's triangle.
- Understand recurrence relations and reason recursively.
- Model situations with networks and use graph theory to solve problems.
- Demonstrate familiarity with algorithms and their implementation and efficiencies.
- Identify and analyze common sequences and their characteristics (e.g., Fibonacci, triangular numbers, arithmetic, geometric progressions).
- Apply concepts from logic and logical reasoning puzzles and related mathematical problems.
- Understand and appreciate how discrete mathematics is used in real-world situations such as theoretical computer science and cybersecurity.
Subarea 5—Mathematics Practices, Dispositions, Curriculum
Objective 019—Demonstrate Mathematical Practices
Includes:
- Engage in Mathematical Practices (i.e., Make sense of problems and persevere in solving them 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; and Look for and express regularity in repeated reasoning).
- Understand that doing mathematics is a sense-making activity that calls for perseverance, problem posing, and problem-solving.
- Explain mathematical thinking using grade-appropriate concepts, procedures, and language, including grade-appropriate definitions and interpretations for key mathematical concepts.
- Describe metacognitive actions and behaviors used in mathematical thinking.
- Recognize the interrelationships among the Practices and how they support each other, and those that are important to a mathematical investigation.
Objective 020—Exhibit Productive Mathematical Dispositions
Includes:
- Demonstrate knowledge of how a productive disposition of mathematics contributes to success as a practitioner of mathematics.
- Understand that the teacher has a central role in student learning of mathematics and view a teacher's role as supporting the development of students' robust and powerful mathematical identities.
- Describe mathematics as a sense-making activity that calls for habits of mind such as curiosity, imagination, inventiveness, risk-taking, and persistence.
- Demonstrate understanding that mathematics makes sense and is useful and worthwhile, believe that steady effort in learning mathematics pays off, and see oneself as an effective learner and doer of mathematics.
- Identify beliefs and classroom practices that produce equitable and inequitable mathematical learning experiences and outcomes for students and seek to create more equitable learning environments.
- Recognize ways to view oneself as a perpetual learner of mathematics, and demonstrate knowledge of practices for ongoing learning in mathematics, including looking for new and innovative ways to solve problems and seek out new mathematical tools and techniques.
Objective 021—Analyze the Mathematical Content of Curriculum
Includes:
- Understand the content within standards documents, learning progressions, mathematics curricula, instructional materials, and assessment frameworks and be able to discuss them with colleagues, administrators, and families and caregivers of students in ways that make sense to each audience.
- Connect standards documents, learning progressions, mathematics curricula, instructional materials, and assessment frameworks and have the commitment to analyze these guides to inform their teaching.
- Analyze provided instructional resources and formative assessment data to make decisions about the sequencing and time required to teach the content as well as how to make important connections among the mathematics taught in the grades and/or units before and after what is currently being taught.
- Apply knowledge of content and practices to critically analyze multiple mathematical instructional resources, to determine whether resources fully address the content and practice expectations described in standards and curriculum documents, and to promote equitable and effective teaching.
Objective 022—Use Mathematical Tools and Technology
Includes:
- Select and use tools and technology for solving mathematical problems, for mathematical modeling, and for supporting mathematical reasoning and sense-making.
- Select and use manipulatives (e.g., fold paper, toss coins, demonstrate polyhedra models, measure with protractors) and technology (e.g., interact with an applet displaying slope triangles for a line, find the intersection point of two functions using a graphing calculator, drag a point on a number line, change values in a spreadsheet to explore the meaning of variable) as a means of developing students' understanding of mathematics.
- Employ the strategic use of virtual manipulatives and interactive electronic depictions of physical manipulatives, know how these can support sophisticated explorations of mathematical concepts, and make sound decisions about when such tools enhance teaching and learning.
- Understand the benefits of using physical and virtual manipulatives and make strategic choices between them, recognizing their potential and limitations for students' mathematics learning.
- Understand the benefits of staying abreast of new tools and shifts in technology to support students' mathematical learning.