TExMaT Master Mathematics Teacher 8-12 (089) Test at a Glance

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TExMaT™ Master Mathematics Teacher 8–12 (089) Test at a Glance

See the test preparation manual for complete information about the test along with sample questions, study tips and preparation resources.

Test Name Master Mathematics Teacher 8–12

Test Code 089

Time 5 hours

Number of Questions 90 multiple-choice questions 1 case study assignment

Format Paper-based test (PBT)

Domain Domain Title

Approx. Percentage

of Test

I.

Number Concepts: Content, Instruction and Assessment 16%

II.

Patterns and Algebra: Content, Instruction and Assessment 18%

III.

Precalculus and Calculus: Content, Instruction and Assessment 16%

IV.

Geometry and Measurement: Content, Instruction and Assessment 18%

V.

Probability and Statistics: Content, Instruction and Assessment 14%

VI.

Mathematical Processes, Perspectives, Mentoring and Leadership 18%

I. 16%

II. 18%

III. 16%

IV. 18%

V. 14%

VI. 18%

TExMaT Master Mathematics Teacher 8–12 (089) Page 2 of 33

About This Test The TExMaT Master Mathematics Teacher 8–12 (089) test is designed to assess whether an examinee has the requisite knowledge and skills that an initially certified Texas Master Mathematics Teacher 8–12 must possess. The 90 multiple-choice questions are based on the Master Mathematics Teacher 8–12 test framework. Questions on this test range from grades 8–12. The test may contain questions that do not count toward the score.

The Test Framework The Master Mathematics Teacher 8–12 test framework is based on the educator standards for this field. The content covered by the test is organized into broad areas of content called domains. Each domain covers one or more of the educator standards for this field. Within each domain, the content is further defined by a set of competencies. Each competency is composed of two major parts: The competency statement, which broadly defines what an entry-level

educator in this field in Texas public schools should know and be able to do.

The descriptive statements, which describe in greater detail the knowledge and skills eligible for testing.

The educator standards being assessed within each domain are listed beginning on the next page. These are followed by a complete set of the framework’s competencies and descriptive statements. Read each competency with its descriptive statements to get a more specific idea of the knowledge you will be required to demonstrate on the test.


Educator Standards Master Mathematics Teacher 8–12 Standard I Number Concepts: The Master Mathematics Teacher understands and applies knowledge of numbers, number systems and their structure, operations and algorithms, quantitative reasoning and the vertical alignment of number concepts to teach the statewide curriculum (Texas Essential Knowledge and Skills [TEKS]). Master Mathematics Teacher 8–12 Standard II Patterns and Algebra: The Master Mathematics Teacher understands and applies knowledge of patterns, relations, functions, algebraic reasoning, analysis and the vertical alignment of patterns and algebra to teach the statewide curriculum (Texas Essential Knowledge and Skills [TEKS]). Master Mathematics Teacher 8–12 Standard III Geometry and Measurement: The Master Mathematics Teacher understands geometry, spatial reasoning, measurement concepts and principles and the vertical alignment of geometry and measurement to teach the statewide curriculum (Texas Essential Knowledge and Skills [TEKS]). Master Mathematics Teacher 8–12 Standard IV Probability and Statistics: The Master Mathematics Teacher understands probability and statistics, their applications and the vertical alignment of probability and statistics to teach the statewide curriculum (Texas Essential Knowledge and Skills [TEKS]). Master Mathematics Teacher 8–12 Standard V Mathematical Processes: The Master Mathematics Teacher understands and uses mathematical processes to reason mathematically, to solve mathematical problems, to make mathematical connections within and outside of mathematics and to communicate mathematically. Master Mathematics Teacher 8–12 Standard VI Instruction: The Master Mathematics Teacher applies knowledge of mathematical content, uses appropriate theories for learning mathematics, implements effective instructional approaches for teaching mathematics, including teaching students who are at-risk, and demonstrates effective classroom management techniques. Master Mathematics Teacher 8–12 Standard VII Creating and Promoting a Positive Learning Environment: The Master Mathematics Teacher demonstrates behavior that reflects high expectations for every student, promotes positive student attitudes towards mathematics and provides equitable opportunities for all students to achieve at a high level. Master Mathematics Teacher 8–12 Standard VIII Assessment: The Master Mathematics Teacher selects, constructs and administers appropriate assessments to guide, monitor, evaluate and report student progress to students, administrators and parents, and develops these skills in other teachers.


Master Mathematics Teacher 8–12 Standard IX Mentoring and Leadership: The Master Mathematics Teacher facilitates appropriate standards-based mathematics instruction by communicating and collaborating with educational stake-holders; mentoring, coaching, exhibiting leadership and consulting with colleagues; providing professional development opportunities for faculty; and making instructional decisions based on data and supported by evidence from research. Master Mathematics Teacher 8–12 Standard X Mathematical Perspectives: The Master Mathematics Teacher understands the historical development of mathematical ideas, the interrelationship between society and mathematics, the structure of mathematics and the evolving nature of mathematics and mathematical knowledge.


Domains and Competencies

DOMAIN I — NUMBER CONCEPTS: CONTENT, INSTRUCTION AND ASSESSMENT

Standards Assessed: Master Mathematics Teacher 8–12 I, VI–VIII

Competency 001: The Master Mathematics Teacher 8–12 understands the real number system and its structure, operations, algorithms and representations.

The Master Mathematics Teacher 8–12: A. Understands the concepts of place value and decimal representations of

real numbers.

B. Understands the algebraic structure of the real number system and its subsets (e.g., real numbers as a field, integers as an additive group).

C. Describes and analyzes properties of subsets of the real numbers (e.g., closure, identities).

D. Selects and uses appropriate representations of real numbers for particular situations.

E. Uses a variety of models (e.g., concrete, pictorial, geometric, symbolic) to represent operations, algorithms and real numbers.

F. Uses real numbers to model and solve a variety of problems.

G. Uses deductive reasoning to simplify and justify algebraic processes.

H. Demonstrates how some problems that have no solution in the integer or rational number systems have solutions in the real number system.


Competency 002: The Master Mathematics Teacher 8–12 understands the complex number system and its structure, operations, algorithms and representations.

The Master Mathematics Teacher 8–12: A. Understands the algebraic structure of the complex number system

and its subsets (e.g., complex numbers as a field, complex addition as vector addition).

B. Understands the properties and operations of complex numbers (e.g., complex conjugate, magnitude/modulus, multiplicative inverse).

C. Describes and analyzes properties of subsets of the complex numbers (e.g., closure, identities).

D. Selects and uses appropriate representations of complex numbers (e.g., vector, ordered pair, polar, exponential) for particular situations.

E. Describes complex number operations (e.g., addition, multiplication, powers, roots) using symbolic and geometric representations.

F. Demonstrates how some problems that have no solution in the real number system have solutions in the complex number system.

Competency 003: The Master Mathematics Teacher 8–12 understands number theory concepts and principles and uses numbers to model and solve problems in a variety of situations.

The Master Mathematics Teacher 8–12: A. Applies ideas from number theory (e.g., prime numbers and factorization,

the Euclidean algorithm, divisibility, congruence classes, modular arithmetic, the fundamental theorem of arithmetic) to solve problems.

B. Applies number theory concepts and principles to justify and prove number relationships.

C. Compares and contrasts properties of vectors and matrices with properties of number systems (e.g., existence of inverses, noncommutative operations).

D. Uses a variety of manipulatives to represent number properties.

E. Uses properties of numbers (e.g., fractions, decimals, percents, ratios, proportions) to model and solve real-world problems.

F. Applies counting techniques, such as permutations and combinations, to quantify situations and solve problems.

G. Uses estimation techniques to solve problems and evaluates the reasonableness of solutions.


Competency 004: The Master Mathematics Teacher 8–12 plans and designs effective instruction and assessment based on knowledge of how all students, including students who are at-risk, learn and develop number concepts, skills and procedures.

The Master Mathematics Teacher 8–12: A. Evaluates and applies established research evidence on how all students,

including students who are at-risk, learn and use number concepts.

B. Recognizes and uses the vertical alignment of number concepts across grade levels to plan instruction based on state standards.

C. Sequences instruction, practice and applications based on students’ instructional needs so that all students develop accuracy and fluency of number concepts.

D. Uses evidence of students’ current understanding of number concepts to select strategies to help students move from informal to formal knowledge.

E. Structures problem-solving activities so students can recognize patterns and relationships within number concepts.

F. Designs challenging and engaging problem-solving tasks that develop number-concepts content knowledge as well as students’ critical and analytical reasoning capacities.

G. Integrates number concepts within and outside of mathematics.

H. Selects appropriate materials, instructional strategies and technology to meet the instructional needs of all students.

I. Uses strategies to help students understand that results obtained using technology may be misleading and/or misinterpreted.

J. Recognizes common errors and misconceptions and determines appropriate correction procedures.

K. Develops assessments based on state and national standards to evaluate students’ knowledge of number concepts.

L. Evaluates an assessment for validity with respect to the measured objectives.

M. Analyzes and uses assessment results from various diagnostic instruments to plan, inform and adjust instruction.

N. Recognizes how to provide equity for all students in mathematics instruction through reflection on one’s own attitudes, expectations and teaching practices.


Competency 005: The Master Mathematics Teacher 8–12 implements a variety of instruction and assessment techniques to guide, evaluate and improve students’ learning of number concepts, skills and procedures.

The Master Mathematics Teacher 8–12: A. Creates a positive learning environment that provides all students

with opportunities to develop and improve number concepts, skills and procedures.

B. Knows how to teach number concepts, skills, procedures and problem-solving strategies using instructional approaches supported by established research.

C. Knows how to maximize student/teacher and student/student interaction and analyzes students’ abilities to correctly apply new content.

D. Uses multiple representations, tools and a variety of tasks to promote students’ understanding of number concepts.

E. Introduces content by carefully defining new terms using vocabulary that the student already knows.

F. Uses a variety of questioning strategies to identify, support, monitor and challenge students’ mathematical thinking.

G. Demonstrates classroom management skills, including applying strategies that use instructional time effectively.

H. Administers a variety of appropriate assessment instruments and/or methods (e.g., formal/informal, formative/summative) consisting of worthwhile tasks that assess mathematical understanding, common misconceptions and error patterns associated with learning number concepts.

I. Evaluates and modifies instruction to improve learning of number concepts, skills and procedures for all students based on the results of formal and informal assessments.


DOMAIN II — PATTERNS AND ALGEBRA: CONTENT, INSTRUCTION AND ASSESSMENT

Standards Assessed: Master Mathematics Teacher 8–12 II, VI–VIII

Competency 006: The Master Mathematics Teacher 8–12 uses patterns to model and solve problems and formulate conjectures.

The Master Mathematics Teacher 8–12: A. Recognizes, extends and generalizes patterns and relationships in

information presented in a variety of ways using concrete models, geometric figures, tables, graphs and algebraic expressions.

B. Uses methods of recursion and iteration to model and solve problems.

C. Uses the principle of mathematical induction.

D. Analyzes the properties of sequences and series (e.g., Fibonacci, arithmetic, geometric) and uses them to solve problems involving finite and infinite processes.

E. Understands how sequences and series are applied to solve problems in the mathematics of finance (e.g., simple, compound, and continuous interest rates; annuities).


Competency 007: The Master Mathematics Teacher 8–12 understands attributes of functions, relations and their graphs.

The Master Mathematics Teacher 8–12: A. Understands when a relation is a function.

B. Identifies the mathematical domain and range of functions and relations and determines reasonable domains for given situations.

C. Understands that there exist functions that cannot be represented by any mathematical equation.

D. Understands that a function represents a dependence of one quantity on another and can be represented in a variety of ways (e.g., concrete models, tables, graphs, diagrams, verbal descriptions, symbols).

E. Identifies and analyzes even and odd functions, one-to-one functions, inverse functions and their graphs.

F. Applies basic transformations [e.g., k f(x), f(x) + k, f(x – k), f(kx), │f(x)│] to a parent function, f, and describes the effects on the graph of y = f(x).

G. Performs operations (e.g., sum, difference, composition) on functions, finds inverse relations and describes the results using multiple representations (e.g., concrete, verbal, graphic, symbolic).

H. Uses graphs of functions to formulate conjectures of identities [e.g., y = x2 – 1 and y = (x – 1)(x + 1), y = log x3 and y = 3 log x, y = sin(x + ) and y = cos x].

I. Represents and solves problems using parametric and polar equations using a variety of methods, including technology.


Competency 008: The Master Mathematics Teacher 8–12 understands linear functions, analyzes their algebraic and graphical properties and uses them to model and solve problems.

The Master Mathematics Teacher 8–12: A. Understands the relationship between linear models and rate of change.

B. Interprets the meaning of slope and intercept in a variety of situations.

C. Analyzes the relationship between a linear equation and its graph.

D. Writes equations of lines given various characteristics (e.g., two points, a point and slope, slope and y-intercept).

E. Determines the linear function that best models a set of data.

F. Uses a variety of methods (e.g., numeric, algebraic, graphic) to solve problems involving systems of linear equations and inequalities.

G. Applies techniques of linear and matrix algebra to represent and solve problems involving linear systems.

H. Models and solves problems involving linear equations and inequalities using a variety of methods, including technology.

Competency 009: The Master Mathematics Teacher 8–12 understands quadratic functions, analyzes their algebraic and graphical properties and uses them to model and solve problems.

The Master Mathematics Teacher 8–12: A. Manipulates and simplifies quadratic expressions.

B. Analyzes the zeros (real and complex) of quadratic functions.

C. Understands connections and translates among geometric, graphic, numeric and symbolic representations of quadratic functions.

D. Makes connections between the y = ax2 + bx + c and the y = a(x – h)2 + k representations of a quadratic function and its graph.

E. Solves problems involving quadratic functions using a variety of methods (e.g., factoring, completing the square, using the quadratic formula, using a graphing calculator).

F. Models and solves problems involving quadratic equations, inequalities and systems using a variety of methods, including technology.


Competency 010: The Master Mathematics Teacher 8–12 plans and designs effective instruction and assessment based on knowledge of how all students, including students who are at-risk, learn and develop patterns and algebra concepts, skills, and procedures.

The Master Mathematics Teacher 8–12: A. Evaluates and applies established research evidence on how all students,

including students who are at-risk, learn and use patterns and algebra.

B. Recognizes and uses the vertical alignment of patterns and algebra across grade levels to plan instruction based on state standards.

C. Sequences instruction, practice and applications based on students’ instructional needs so that students develop accuracy and fluency of patterns and algebra.

D. Uses evidence of students’ current understanding of patterns and algebra to select strategies to help students move from informal to formal knowledge.

E. Structures problem-solving activities so students can recognize patterns and relationships within patterns and algebra.

F. Designs challenging and engaging problem-solving tasks that develop patterns and algebra content knowledge as well as students’ critical and analytical reasoning capacities.

G. Integrates patterns and algebra concepts within and outside of mathematics.


I. Uses strategies to help students understand that results obtained using technology may be misleading or misinterpreted.


K. Develops assessments based on state and national standards to evaluate students’ knowledge of patterns and algebra.



N. Recognizes how to provide equity for all students in mathematics instruction through reflection on one's own attitudes, expectations and teaching practices.


Competency 011: The Master Mathematics Teacher 8–12 implements a variety of instruction and assessment techniques to guide, evaluate and improve students’ learning of patterns and algebra concepts, skills, and procedures.

The Master Mathematics Teacher 8–12: A. Creates a positive learning environment that provides all students with

opportunities to develop and improve patterns and algebra concepts, skills and procedures.

B. Knows how to teach patterns and algebra concepts, skills, procedures and problem-solving strategies using instructional approaches supported by established research.


D. Uses multiple representations, tools and a variety of tasks to promote students’ understanding of patterns and algebra concepts.




H. Administers a variety of appropriate assessment instruments and/or methods (e.g., formal/informal, formative/summative) consisting of worthwhile tasks that assess mathematical understanding, common misconceptions and error patterns associated with learning patterns and algebra concepts.

I. Evaluates and modifies instruction to improve learning of patterns and algebra concepts, skills and procedures for all students based on the results of formal and informal assessments.


DOMAIN III — PRECALCULUS AND CALCULUS: CONTENT, INSTRUCTION AND ASSESSMENT

Standards Assessed: Master Mathematics Teacher 8–12 II–III, VI–VIII

Competency 012: The Master Mathematics Teacher 8–12 understands polynomial, rational, radical, absolute value and piecewise-defined functions and relations, analyzes their algebraic and graphical properties and uses them to model and solve problems.

The Master Mathematics Teacher 8–12: A. Recognizes and translates among multiple representations (e.g., written,

tabular, graphical, algebraic) of polynomial, rational, radical, absolute value and piecewise-defined functions and relations.

B. Describes restrictions on the domains and ranges of polynomial, rational, radical, absolute value and piecewise-defined functions and relations.

C. Makes and uses connections among the significant points (e.g., zeros, local extrema, points where a function is not continuous or not differentiable) of a function, the graph of the function and the function's symbolic representation.

D. Analyzes functions and relations in terms of vertical, horizontal and slant asymptotes.

E. Analyzes and applies the relationships among inverse variation and rational functions and relations.

F. Solves equations and inequalities involving polynomial, rational, radical, absolute value and piecewise-defined functions and relations using a variety of methods (e.g., tables, algebraic methods, graphs, using a graphing calculator) and evaluates the reasonableness of solutions.

G. Models situations using polynomial, rational, radical, absolute value and piecewise-defined functions and relations and solves problems using a variety of methods, including technology.

H. Identifies and analyzes the relationships between the properties of conic sections (e.g., foci, axes of symmetry, asymptotes, eccentricity) and their appropriate second-degree equation.

I. Explores and solves application problems involving conic sections.


Competency 013: The Master Mathematics Teacher 8–12 understands exponential and logarithmic functions, analyzes their algebraic and graphical properties and uses them to model and solve problems.

The Master Mathematics Teacher 8–12:

A. Recognizes and translates among multiple representations (e.g., written, numerical, tabular, graphical, algebraic) of exponential and logarithmic functions.

B. Recognizes and uses connections among significant characteristics (e.g., intercepts, asymptotes) of a function involving exponential or logarithmic expressions, the graph of the function and the function’s symbolic representation.

C. Understands the relationship between exponential and logarithmic functions and uses the laws and properties of exponents and logarithms to simplify expressions and solve problems.

D. Uses a variety of representations and techniques (e.g., numerical methods, tables, graphs, analytic techniques, graphing calculators) to solve equations, inequalities and systems involving exponential and logarithmic functions.

E. Models and solves problems involving exponential growth and decay.

F. Uses logarithmic scales (e.g., Richter, decibel) to describe phenomena and solve problems.

G. Uses exponential and logarithmic functions to model and solve problems, including problems involving the mathematics of finance (e.g., compound interest).

H. Uses the exponential function to model situations and solve problems in which the rate of change of a quantity is proportional to the current amount of the quantity [i.e., f '(x) = k f(x)].

I. Models and solves problems involving logistic functions.


Competency 014: The Master Mathematics Teacher 8–12 understands trigonometric and circular functions, analyzes their algebraic and graphical properties and uses them to model and solve problems.


A. Analyzes the relationships among the unit circle in the coordinate plane, circular functions and the trigonometric functions.

B. Solves problems using angular- and linear-velocity concepts.

C. Recognizes and translates among multiple representations (e.g., written, numerical, tabular, graphical, algebraic) of trigonometric functions and their inverses.

D. Relates trigonometry and algebra by expressing equations in rectangular, parametric and polar representations.

E. Solves problems using the law of sines and the law of cosines.

F. Recognizes and uses connections among significant properties (e.g., zeros, axes of symmetry, local extrema) and characteristics (e.g., amplitude, frequency, phase shift) of a trigonometric function, the graph of the function and the function’s symbolic representation.

G. Understands the relationships between trigonometric functions and their inverses and uses these relationships to solve problems.

H. Uses trigonometric identities to simplify expressions and solve equations.

I. Models and solves a variety of problems (e.g., analyzing periodic phenomena) using trigonometric functions.

J. Uses technology to analyze and solve problems involving trigonometric functions.


Competency 015: The Master Mathematics Teacher 8–12 understands and solves problems using limits, continuity and differential calculus.


A. Understands and applies the concepts of limit, continuity and differentiability.

B. Understands the definition and properties of the derivative.

C. Relates the concept of average rate of change to the slope of the secant line and the concept of instantaneous rate of change to the slope of the tangent line.

D. Understands and applies techniques of differentiation (e.g., product rule, chain rule).

E. Uses the first and second derivatives to analyze the graph of a function (e.g., local extrema, concavity, points of inflection).

F. Models and solves a variety of problems (e.g., velocity, acceleration, optimization, related rates) using differential calculus.

G. Analyzes how technology can be used to solve problems and illustrate concepts involving differential calculus.

Competency 016: The Master Mathematics Teacher 8–12 understands and solves problems using integral calculus.


A. Understands and applies the fundamental theorem of calculus and the relationship between differentiation and integration.

B. Relates the concept of area under a curve to the limit of a Riemann sum.

C. Understands and applies techniques of integration.

D. Uses integral calculus to compute various measurements (e.g., area, volume, arc length) associated with curves and regions in the plane, and measurements associated with curves, surfaces and regions in three-dimensional space.

E. Models and solves a variety of problems (e.g., displacement, velocity, work, center of mass) using integral calculus.

F. Analyzes how technology can be used to solve problems and illustrate concepts involving integral calculus.


Competency 017: The Master Mathematics Teacher 8–12 plans and designs effective instruction and assessment based on knowledge of how all students, including students who are at-risk, learn and develop precalculus and calculus concepts, skills and procedures.


A. Evaluates and applies established research evidence on how all students, including students who are at-risk, learn and use precalculus and calculus.

B. Recognizes and uses the vertical alignment of precalculus and calculus across grade levels to plan instruction based on state standards.

C. Sequences instruction, practice and applications based on students’ instructional needs so that all students develop accuracy and fluency of precalculus and calculus.

D. Uses evidence of students’ current understanding of precalculus and calculus to select strategies to help students move from informal to formal knowledge.

E. Structures problem-solving activities so students can recognize patterns and relationships within precalculus and calculus.

F. Designs challenging and engaging problem-solving tasks that develop precalculus and calculus content knowledge as well as students’ critical and analytical reasoning capacities.

G. Integrates precalculus and calculus within and outside of mathematics.




K. Develops assessments based on state and national standards to evaluate students’ knowledge of precalculus and calculus.





Competency 018: The Master Mathematics Teacher 8–12 implements a variety of instruction and assessment techniques to guide, evaluate and improve student learning of precalculus and calculus concepts, skills and procedures.


A. Creates a positive learning environment that provides all students with opportunities to develop and improve precalculus and calculus concepts, skills and procedures.

B. Knows how to teach precalculus and calculus concepts, skills, procedures and problem-solving strategies using instructional approaches supported by established research.


D. Uses multiple representations, tools and a variety of tasks to promote students’ understanding of precalculus and calculus.




H. Administers a variety of appropriate assessment instruments and/or methods (e.g., formal/informal, formative/summative) consisting of worthwhile tasks that assess mathematical understanding, common misconceptions and error patterns associated with learning precalculus and calculus.

I. Evaluates and modifies instruction to improve learning of precalculus and calculus concepts, skills and procedures for all students based on the results of formal and informal assessments.


DOMAIN IV — GEOMETRY AND MEASUREMENT: CONTENT, INSTRUCTION AND ASSESSMENT

Standards Assessed: Master Mathematics Teacher 8–12 III, VI–VIII

Competency 019: The Master Mathematics Teacher 8–12 understands measurement as a process.


A. Applies dimensional analysis to derive units and formulas in a variety of situations (e.g., rates of change of one variable with respect to another) and to find and evaluate solutions to problems.

B. Applies formulas for perimeter, area, surface area and volume of geometric shapes and solids (e.g., polygons, pyramids, prisms, cylinders, cones, spheres) to solve problems.

C. Solves problems involving capacity, mass, weight, density, time, temperature, angles and rates of change.

D. Recognizes the effects on length, area or volume when linear dimensions are changed.

E. Applies the Pythagorean theorem, proportional reasoning and right triangle trigonometry to solve measurement problems.

F. Uses methods of approximation and estimation and understands the effects of error on measurement.


Competency 020: The Master Mathematics Teacher 8–12 understands geometries, in particular Euclidean geometry, as axiomatic systems.


A. Understands axiomatic systems and their components (e.g., undefined terms, defined terms, theorems, examples, counterexamples).

B. Uses properties of points, lines, planes, angles, lengths and distances to solve problems.

C. Applies the properties of parallel and perpendicular lines to solve problems.

D. Uses properties of congruence and similarity to explore geometric relationships, justify conjectures and prove theorems.

E. Describes and justifies geometric constructions made using compass and straightedge, reflection devices and other appropriate technologies.

F. Demonstrates an understanding of the use of appropriate software to explore attributes of geometric figures and to make and evaluate conjectures about geometric relationships.

G. Compares and contrasts the axioms of Euclidean geometry with those of non-Euclidean geometry.

H. Demonstrates an understanding of proof, including indirect proof, in geometry, and provides convincing arguments or proofs for geometric theorems.


Competency 021: The Master Mathematics Teacher 8–12 understands the results, uses and applications of Euclidean geometry.


A. Analyzes the properties of polygons and their components.

B. Analyzes the properties of circles and the lines that intersect them.

C. Uses geometric patterns and properties (e.g., similarity, congruence) to make generalizations about two- and three-dimensional figures (e.g., relationships of sides, angles).

D. Computes the perimeter, area and volume of shapes and solids created by subdividing and combining other shapes and solids (e.g., arc length, area of sectors, volume of a hemisphere).

E. Analyzes cross sections and nets of three-dimensional solids.

F. Uses top, front, side and corner views of three-dimensional solids to create complete representations and solve problems.

G. Applies properties of two- and three-dimensional figures to solve problems in other disciplines and in everyday life.

Competency 022: The Master Mathematics Teacher 8–12 understands coordinate, transformational and vector geometry and their connections.


A. Identifies transformations (i.e., reflections, translations, rotations, dilations) and explores their properties.

B. Uses the properties of transformations and their compositions to solve problems.

C. Uses transformations to explore and describe reflectional, rotational and translational symmetry.

D. Applies transformations in the coordinate plane.

E. Applies concepts and properties of slope, midpoint, parallelism, perpendicularity and distance to explore properties of geometric figures and solve problems in the coordinate plane.

F. Uses rectangular and polar coordinate geometry to derive and explore the equations, properties and applications of conic sections, including degenerate conic sections.

G. Relates geometry and algebra by representing transformations as matrices and uses this relationship to solve problems.

H. Explores the relationship between geometric and algebraic representations of vectors and uses this relationship to solve problems.


Competency 023: The Master Mathematics Teacher 8–12 plans and designs effective instruction and assessment based on knowledge of how all students, including students who are at-risk, learn and develop geometry and measurement concepts, skills, and procedures.


A. Evaluates and applies established research evidence on how all students, including students who are at-risk, learn and use geometry and measurement.

B. Recognizes and uses the vertical alignment of geometry and measurement across grade levels to plan instruction based on state standards.

C. Sequences instruction, practice and applications based on students’ instructional needs so that all students develop accuracy and fluency of geometry and measurement.

D. Uses evidence of students’ current understanding of geometry and measurement to select strategies to help students move from informal to formal knowledge.

E. Structures problem-solving activities so students can recognize patterns and relationships within geometry and measurement.

F. Designs challenging and engaging problem-solving tasks that develop geometry and measurement content knowledge as well as students’ critical and analytical reasoning capacities.

G. Integrates geometry and measurement within and outside of mathematics.




K. Develops assessments based on state and national standards to evaluate students’ knowledge of geometry and measurement.





Competency 024: The Master Mathematics Teacher 8–12 implements a variety of instruction and assessment techniques to guide, evaluate and improve students’ learning of geometry and measurement concepts, skills and procedures.


A. Creates a positive learning environment that provides all students with opportunities to develop and improve geometry and measurement concepts, skills and procedures.

B. Knows how to teach geometry and measurement concepts, skills, procedures and problem-solving strategies using instructional approaches supported by established research.


D. Uses multiple representations, tools and a variety of tasks to promote students’ understanding of geometry and measurement.




H. Administers a variety of appropriate assessment instruments and/or methods (e.g., formal/informal, formative/summative) consisting of worthwhile tasks that assess mathematical understanding, common misconceptions and error patterns associated with learning geometry and measurement.

I. Evaluates and modifies instruction to improve learning of geometry and measurement concepts, skills and procedures for all students based on the results of formal and informal assessments.


DOMAIN V — PROBABILITY AND STATISTICS: CONTENT, INSTRUCTION AND ASSESSMENT

Standards Assessed: Master Mathematics Teacher 8–12 IV, VI–VIII

Competency 025: The Master Mathematics Teacher 8–12 understands how to use appropriate graphical and numerical techniques, including the use of technology, to explore, analyze and represent data.


A. Selects and uses an appropriate measurement scale (i.e., nominal, ordinal, interval, and ratio) to answer research questions and analyze data.

B. Organizes, displays and interprets data in a variety of formats (e.g., tables, frequency distributions, scatterplots, stem-and-leaf plots, box-and-whisker plots, histograms, pie charts).

C. Applies concepts of center, spread, shape and skewness to describe a data distribution.

D. Understands measures of central tendency (i.e., mean, median and mode) and dispersion (i.e., range, interquartile range, variance and standard deviation).

E. Applies linear transformations to convert data and describes the effects of linear transformations on measures of central tendency and dispersion.

F. Analyzes connections among concepts of center and spread, data clusters and gaps, data outliers and measures of central tendency and dispersion.

G. Supports arguments, makes predictions and draws conclusions using summary statistics and graphs to analyze and interpret univariate data.

H. Recognizes and uses appropriate graphical displays and descriptive statistics for categorical and numerical data.


Competency 026: The Master Mathematics Teacher 8–12 understands concepts and applications of probability.


A. Understands how to explore concepts of probability through sampling, experiments and simulations.

B. Generates and uses probability models to represent situations.

C. Uses the concepts and principles of probability to describe the outcomes of simple and compound events.

D. Determines probabilities by constructing sample spaces to model situations.

E. Solves a variety of probability problems using combinations and permutations.

F. Solves a variety of probability problems using ratios of areas of geometric regions.

G. Calculates probabilities using the axioms of probability and related theorems and concepts, such as the addition rule, multiplication rule and conditional probability.

H. Applies concepts and properties of discrete and continuous random variables to model and solve a variety of problems involving probability and probability distributions (e.g., binomial, geometric, uniform, normal).

I. Understands expected value, variance and standard deviation of probability distributions (e.g., binomial, geometric, uniform, normal).


Competency 027: The Master Mathematics Teacher 8–12 understands the relationships among probability theory, sampling and statistical inference, and how statistical inference is used in making and evaluating predictions.


A. Applies knowledge of designing, conducting, analyzing and interpreting statistical experiments to investigate real-world problems.

B. Analyzes and interprets statistical information (e.g., the results of polls and surveys) and recognizes misleading as well as valid uses of statistics.

C. Understands random samples and sample statistics (e.g., sample size, confidence intervals, biased or unbiased estimators).

D. Makes inferences about a population using binomial, normal and geometric distributions.

E. Describes and analyzes bivariate data using various techniques (e.g., scatterplots, regression lines, outliers, residual analysis, correlation coefficients).

F. Understands how to transform nonlinear data into linear form in order to apply linear regression techniques to develop exponential, logarithmic and power regression models.

G. Understands the law of large numbers and the central limit theorem and their connections to the process of statistical inference.

H. Estimates parameters (e.g., population mean and variance) using point estimators (e.g., sample mean and variance).

I. Understands principles of hypothesis testing.

J. Analyzes categorical data (e.g., frequency tables, chi-square).


Competency 028: The Master Mathematics Teacher 8–12 plans and designs effective instruction and assessment based on knowledge of how all students, including students who are at-risk, learn and develop probability and statistics concepts, skills, and procedures.


A. Evaluates and applies established research evidence on how all students, including students who are at-risk, learn and use probability and statistics.

B. Recognizes and uses the vertical alignment of probability and statistics across grade levels to plan instruction based on state standards.

C. Sequences instruction, practice and applications based on students’ instructional needs so that all students develop accuracy and fluency of probability and statistics.

D. Uses evidence of students’ current understanding of probability and statistics to select strategies to help students move from informal to formal knowledge.

E. Structures problem-solving activities so students can recognize patterns and relationships within probability and statistics.

F. Designs challenging and engaging problem-solving tasks that develop probability and statistics content knowledge as well as students’ critical and analytical reasoning capacities.

G. Integrates probability and statistics within and outside of mathematics.




K. Develops assessments based on state and national standards to evaluate students’ knowledge of probability and statistics.





Competency 029: The Master Mathematics Teacher 8–12 implements a variety of instruction and assessment techniques to guide, evaluate and improve students’ learning of probability and statistics concepts, skills and procedures.


A. Creates a positive learning environment that provides all students with opportunities to develop and improve probability and statistics concepts, skills and procedures.

B. Knows how to teach probability and statistics concepts, skills, procedures and problem-solving strategies using instructional approaches supported by established research.


D. Uses multiple representations, tools and a variety of tasks to promote students’ understanding of probability and statistics.




H. Administers a variety of appropriate assessment instruments and/or methods (e.g., formal/informal, formative/summative) consisting of worthwhile tasks that assess mathematical understanding, common misconceptions and error patterns associated with learning probability and statistics.

I. Evaluates and modifies instruction to improve learning of probability and statistics concepts, skills and procedures for all students based on the results of formal and informal assessments.


DOMAIN VI — MATHEMATICAL PROCESSES, PERSPECTIVES, MENTORING AND LEADERSHIP

Standards Assessed: Master Mathematics Teacher 8–12 V, IX–X

Competency 030: The Master Mathematics Teacher 8–12 understands and uses mathematical processes to reason mathematically and solve problems.


A. Demonstrates an understanding of the use of logical reasoning to evaluate mathematical conjectures and justifications and to provide convincing arguments or proofs for mathematical theorems.

B. Applies correct mathematical reasoning to derive valid conclusions from a set of premises, and recognizes examples of fallacious reasoning.

C. Demonstrates an understanding of the use of inductive reasoning to make conjectures and deductive methods to evaluate the validity of conjectures.

D. Applies knowledge of the use of formal and informal reasoning to explore, investigate and justify mathematical ideas.

E. Recognizes that a mathematical problem can be solved in a variety of ways and selects an appropriate strategy for a given problem.

F. Evaluates the reasonableness of a solution to a given problem.

G. Demonstrates an understanding of estimation and evaluates its appropriate uses.

H. Uses physical and numerical models to represent a given problem or mathematical procedure.

I. Recognizes that assumptions are made when solving problems; then identifies and evaluates those assumptions.

J. Investigates and explores problems that have multiple solutions.

K. Applies content knowledge to develop a mathematical model of a real-world situation; then analyzes and evaluates how well the model represents the situation.

L. Develops and uses simulations as a tool to model and solve problems.


Competency 031: The Master Mathematics Teacher 8–12 understands mathematical connections, the structure of mathematics, the historical development of mathematics and how to communicate mathematical ideas and concepts.


A. Recognizes and uses multiple representations of a mathematical concept.

B. Uses mathematics to model and solve problems in other disciplines.

C. Uses the structure of mathematical systems and their properties (e.g., mappings, inverse operations) to make connections among mathematical concepts.

D. Recognizes the impacts of technological advances on mathematics (e.g., numerical versus analytical solutions) and of mathematics on technology (binary arithmetic).

E. Emphasizes the role of mathematics in various careers and professions (e.g., economics, engineering) and how technology (e.g., spreadsheets, statistical software) affects the use of mathematics in various careers.

F. Knows and uses the history and evolution of mathematical concepts, procedures and ideas (e.g., the development of non-Euclidean geometry).

G. Recognizes the contributions that different cultures have made to the field of mathematics.

H. Uses current and professional resources to plan and develop activities that provide cultural, historical and technological instruction for the classroom and that connect society and mathematics.

I. Expresses mathematical statements using developmentally appropriate language, standard English, mathematical language and symbolic mathematics.

J. Communicates mathematical ideas using a wide range of technological tools and a variety of representations (e.g., numeric, verbal, graphic, pictorial, symbolic, concrete).

K. Demonstrates an understanding of the use of visual media such as graphs, tables, diagrams and animations to communicate mathematical information.

L. Uses the language of mathematics as a precise means of expressing mathematical ideas.


Competency 032: The Master Mathematics Teacher 8–12 knows how to communicate and collaborate with educational stakeholders to facilitate implementation of appropriate, standards-based mathematics instruction.


A. Knows the dual role of the Master Mathematics Teacher as teacher and mentor in the school community.

B. Knows leadership, communication and facilitation skills and strategies.

C. Knows and applies principles, guidelines and professional ethical standards regarding collegial and professional collaborations, including issues related to confidentiality.

D. Understands the importance of collaborating with administrators, colleagues, parents/guardians and other members of the school community to establish and implement the roles of the Master Mathematics Teacher and ensure effective ongoing communication.

E. Knows strategies for communicating effectively with stakeholders, including other teachers, about using programs and instructional techniques that are based on established research that supports their effectiveness with a range of students, including students who are at-risk.

F. Knows strategies for building trust and a spirit of collaboration with other members of the school community to effect positive change in the school mathematics program and mathematics instruction.

G. Knows how to use leadership skills to ensure the effectiveness and ongoing improvement of the school mathematics program, encourage support for the program and engage others in improving the program.

H. Knows strategies for collaborating with members of the school community to evaluate, negotiate and establish priorities regarding the mathematics program and to facilitate mentoring, professional development and parent/guardian training.

I. Knows strategies for conferring with students, colleagues, administrators and parents/guardians to discuss mathematics-related issues.

J. Knows strategies for collaborating with teachers, administrators and others to identify professional development needs, generate support for professional development programs and ensure provision of effective professional development opportunities.


Competency 033: The Master Mathematics Teacher 8–12 knows how to provide professional development through mentoring, coaching and consultation with colleagues to facilitate implementation of appropriate, standards-based mathematics instruction, and makes instructional decisions supported by established research.


A. Knows and applies skills and strategies for mentoring, coaching and consultation in the development, implementation and evaluation of an effective mathematics program.

B. Knows learning processes and procedures for facilitating adult learning.

C. Knows strategies for facilitating positive change in instructional practices through professional development, mentoring, coaching and consultation.

D. Knows models and features of effective professional development programs that promote sustained applications in classroom practice (e.g., modeling, coaching, follow-up).

E. Knows differences between consultation and supervision.

F. Knows how to use mentoring, coaching and consultation to facilitate team building for promoting student development in mathematics.

G. Knows how to select and use strategies for collaborating with colleagues to identify needs related to mathematics instruction.

H. Knows strategies for collaborating effectively with colleagues with varying levels of skill and experience and/or diverse philosophical approaches to mathematics instruction to develop, implement and monitor mathematics programs.

I. Knows how to select and use strategies to maximize effectiveness as a Master Mathematics Teacher, such as applying principles of time management and engaging in continuous self-assessment.

J. Knows sources for locating information about established research on mathematics learning and understands methods and criteria for reviewing research on mathematics learning.

K. Knows how to critically examine established research on mathematics learning, analyzes its usefulness for addressing instructional needs and applies appropriate procedures for translating research on mathematics learning into practice.

TExMaT Master Mathematics Teacher 8-12 (089) Test at a Glance

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