Prospective mathematics teachers may be required to complete a PRAXIS examination in mathematics as part of their state licensing process. PRAXIS content examinations in mathematics, which have been adopted as assessment tools by many boards of education, are designed for teacher candidates that are seeking licensure as math educators.

The two most widely used examinations for the purpose of qualifying middle and secondary level math teachers include:

- PRAXIS Mathematics: Content Knowledge
- PRAXIS Middle School Mathematics

## PRAXIS Mathematics: Content Knowledge

The PRAXIS Mathematics: Content Knowledge examination consists of 50 multiple-choice questions that may be delivered as a paper examination (0061) or a computer-based examination (5061). Candidates have two hours to complete the examination. The examination’s content categories (and their percentage of the examination) include:

- Algebra and Number Theory: 16 percent
- Calculus: 12 percent
- Data Analysis and Statistics: 10 to 12 percent
- Discrete Mathematics: 6 to 8 percent
- Functions: 16 percent
- Geometry: 10 percent
- Matrix Algebra: 8 to 10 percent
- Measurement: 6 percent
- Probability: 4 to 6 percent
- Trigonometry: 8 percent

The process categories of this PRAXIS II examination include:

- Mathematical connections
- Mathematical problem solving
- Mathematical reasoning and proof

*Algebra Number and Theory*

Students in this content category must be able to demonstrate an understanding of the structure of the natural, real, integer, and complex number systems and perform basic operations within these systems. Students must also be knowledgeable about demonstrating an understanding of the properties of counting numbers; solving ratio, average, percentage and proportion problems; and working with algebraic expressions, equations, and formulas.

*Measurement*

The measurement category reflects the candidate’s ability to make decisions about units and scales within problem situations involving measurement; analyze precision, approximate error and accuracy in measurement situations; and apply informal concepts of successive approximation, limit in measurement situations, and upper and lower bounds.

*Geometry*

Geometry assessment involves understanding how to solve problems using the parts of geometric figures and in two and three dimensions. It also requires candidates to describe relationships among sets of special quadrilaterals, such as the rectangle, square, rhombus, and trapezoid, and solve problems using the properties of circles and those involving inscribed angles.

*Trigonometry*

The PRAXIS II content category of trigonometry involves being able to define and use the six functions of trigonometry using degree or radian measure of angles and understanding their graphs. It also involves applying the formulas for the trigonometric identities and solving trigonometric equations and inequalities.

*Calculus*

Students must demonstrate that they understand what is means for a function to have a limit at a point, and they must also be able to solve problems using the properties of limit. They must be able to show that a particular function is continuous and be able to show they understand the relationship between continuity and differentiability and numerically appropriate derivatives and integrals and show the use standard differentiation and integration techniques.

*Functions*

This content category involves being able to demonstrate an understanding of how to work with functions in various representations and find appropriate family of functions as to model specific phenomena . Students must be able to determine properties of functions and their graphs, determine the composition of two functions, and find the inverse of a one-to-one function.

*Data Analysis and Statistics*

Test takers within this content category must be able to organize data into a suitable form; choose and apply appropriate measures of central tendency and dispersion to describe and compare data sets; and analyze data from specific situations to determine which type of function would most likely model a specific phenomenon.

*Matrix Algebra*

Students must be able to understand vectors and matrixes as systems that have some of the same properties as the real number system; use matrix techniques to solve systems of linear equations; use determinants to reason about inverse of matrices and solutions to systems of equations; and understand and represent translations, rotations, reflections, and dilations of objects in the plane through the use of coordinates, vectors, sketches, and matrixes.

*Discrete Mathematics*

A demonstration of discrete mathematics involves solving basic problems that include counting techniques and using counting techniques to understand various situations. Students must be able to determine a binary relation on a set of symmetric, reflective, or transitive and whether a relation is an equivalence relation. They must also be able to use finite and infinite arithmetic and geometric sequences and series to model simple phenomena.

*Probability*

The probability content category involves understanding the concepts of sample space and probability distribution and understanding the concepts of conditional probability and independent events. They must be able to compute and interpret the expected value of random variables in simple cases and use simulations to construct empirical probability distributions as to make informal inferences about the theoretical probability distribution.

More information on the PRAXIS Mathematics: Content Knowledge examination can be found on the PRAXIS Educational Testing Service website.

## PRAXIS Middle School Mathematics

The PRAXIS Middle School Mathematics examination consists of 40 multiple-choice questions and three, short, constructed-response questions. Teacher candidates are given two hours to take this examination, which is offered as a paper-delivered examination (0069) or a computer-delivered examination (5169).

The PRAXIS Middle School Mathematics exam consists of the following content categories (and their percentage of the examination):

- Arithmetic and Basic Algebra: 20 percent
- Data, Probability, and Statistical Concepts; Discrete Mathematics: 17 percent
- Functions and their Graphs: 13 percent
- Geometry and Measurement: 17 percent
- Problem Solving Exercises (constructed response): 33 percent

The examination’s process categories include:

- Mathematical connections
- Mathematical problem solving
- Mathematical reasoning and proof
- Mathematical representation
- Use of technology

*Arithmetic and Basic Algebra*

Students must be able to identify their ability to add, subtract, multiply, and divide rational numbers expressed in various forms; order any finite set of real numbers and recognize equivalent forms of a number; demonstrate an undestanding of concepts associated with counting numbers; and identify an inverse and the additive and multiplicative inverses of a number.

*Geometry and Measurement*

Test takers must be able to solve problems that involve measurement; compute perimeter and area of a number of figures; compute the surface area and volume of a number of figures; apply the Pythagorean Theorem to solve problems; and user relationships like congruency and similarity to solve problems involving two- and three-dimensional figures.

*Functions and their Graphs*

This content category involves understanding the function notation for functions of one variable and working with the algebraic definition of a function; selecting an equation that best represents a given graph and show an understanding between an equation and its graph; and determining the graphical properties and sketch a graph of a quadratic, linear, step, absolute value, or exponential function.

*Data, Probability, and Statistical Concepts; Discrete Mathematics*

Test takers in this content category must be able to organize data into a presentation that is appropriate for solving a problem; read and analyze date presented in various forms; solve probability problems using finite sample spaces by counting outcomes; and solve problems involving average, such as arithmetic mean and weight averaged.

More information on the PRAXIS Middle School Mathematics examination can be found on the PRAXIS Educational Testing Service website.