135 TExES Mathematics 8 - 12 Exam:The TExES Mathematics 8 -12 Exam is a certification examination that is designed to determine if an individual has the knowledge necessary to teach mathematics at the high school level in the Texas public school system. This exam assesses the individual's knowledge of a variety of mathematical topics and his or her understanding of the teaching methods that are required to effectively teach those topics. This exam is required in order to become a certified mathematics teacher at the high school level in the state of Texas. The exam consists of 90 multiple-choice questions, 80 of which are scored and 10 that are not scored, that are related to the following areas:
- Number Concepts (11 questions)
- Patterns and Algebra (27 questions)
- Geometry and Measurement (15 questions)
- Probability and Statistics (11 questions)
- Mathematical Processes and Perspectives (8 questions)
- Methods of Mathematics Instruction and Assessment (8 questions)
Sample Study Notes
1. Explain what math is and why the basics are important.
Math explains the logic of and relationship between numbers. It is used everyday in countless ways and in order to minimize potential math phobia, teachers need to make the subject relevant to the students' lives and use examples with which they are familiar and that make sense to them. In order to do that, learning the basics is critical, because all math concepts are built on addition, division, fractions and shapes; all mathematical relationships flow from these concepts. It is imperative students understand one concept before moving on to the next. If they fail to grasp the basics, students become confused as they progress to higher levels, because they are unable to apply appropriate background knowledge when introduced to geometry, algebra, probability and statistics. Making math fun by injecting a sense of wonder and excitement into learning how to use numbers in everyday life goes a long way in preventing a fear of math from developing. Some fun activities: play a game of cards, checkers or backgammon; balance a checkbook; or write a rap song.
2. Define mathematics, arithmetic, algebra, geometry, probability, statistics, trigonometry, calculus.
The American Heritage College Dictionary defines mathematics as "the study of the measurement, properties and relationships of quantities using numbers and symbols." It is a formal science of structure, order and relationship, and is considered the basic language and foundation of all the other sciences. It is critical in the development of technology. It evolved from counting, measuring and describing shapes. Some areas of mathematics and their definitions:
- ARITHMETIC: a system to count numbers using addition, subtraction, multiplication and division.
- ALGEBRA: an abstract form of arithmetic using symbols to represent numbers.
- GEOMETRY: the relationship of points, lines, angles, surfaces and solids.
- PROBABILITY: the calculation of the chances that certain events will occur.
- STATISTICS: the collection, organization and analysis of data.
- TRIGONOMETRY: the relationship of the sides and angles of triangles.
- CALCULUS: the limits, differentiation and integration of the functions of variables.
3. Define number concepts.
Number concepts are the building blocks of all mathematical calculations and representations. Students must understand what a number means, how it can and cannot be used, and its relationship to other numbers. They need to be able to depict numbers concretely, pictorially and symbolically. Students need to understand the basic definitions of number concepts in order to use numbers properly in whatever math discipline they are working. These definitions of some common math terms are from The American Heritage College Dictionary.
- INTEGERS are the positive and negative whole numbers and zero
- NATURAL NUMBERS or COUNTING NUMBERS are the positive integers.
- FRACTIONS are the result of dividing one quantity by another quantity.
- A PRIME NUMBER is any number which is only divisible by one (1) and itself.
- A PERCENTAGE is a fraction or ratio expressed as part of one hundred (100).
- A RATIO is the relation between two quantities expressed as the result (quotient) of one divided by the other.
- PLACE VALUE is the position of a figure in a numeral or series.
4. Discuss patterns and functions and equivalence and equations and what they represent.
There are some basic concepts students need to understand in order to begin to think algebraically, so they can use what they "see" to make generalizations about "unknowns." Patterns and functions represent change and relationships. Repeating patterns show the same unit over and over again; in growth patterns, each unit is dependent upon the one before it, as well as its position in the pattern. The function is the relationship between values, e.g. the second depends on the first. Using concrete examples helps students visualize what the function is describing. As students begin to understand functional relationships, symbols can be used as abstract stand-ins for the relationships. Equivalence and balance are critical concepts in understanding algebraic equations. It is important for teachers to explain that the equal sign represents some type of relationship between the numbers and symbols on each side of the sign: if a calculation is performed on one side, the same calculation must be performed on the other side. Each side is equal; the equation must balance.
5. Define geometry and discuss some of its practical applications.
The American Heritage College Dictionary defines geometry as investigation of "properties, measurements and relationships of points, lines, angles, surfaces and solids." Geometry developed from a practical need to determine land boundaries (survey), figure the size (area) of a field, measure the volume of a silo (cylinder), and determine the relative positions of three-dimensional objects in a defined space. Man's fascination with the stars and the heavens became the science of astronomy, which led to the development of trigonometry and its unique computational methods. Studying geometry helps students hone their spatial visualization skills, which helps them function better in the physical world. Points, lines, angles, surfaces and solids are all used in painting, sculpture and architecture. The artist must understand the relationship of these components in order to create in any medium. Various engineering disciplines use geometry to build bridges and dams, design freeway systems, mine for minerals and drill for oil. Geometry is used every day in many professions. Citing real-life examples makes the subject relevant to students' lives outside the classroom.
6. Discuss the importance of statistics.
Statistics is the collection, organization and interpretation of data. The data can be facts or isolated bits of information, but it all relates in one way or another to a specific topic. This precise, analytical system is used to identify, study, and solve problems in many industries. Statistics can help people interpret events and make decisions in uncertain and difficult situations. Healthcare professionals, financial analysts, scientists, engineers and insurance actuaries all use statistics to infer relationships, measure interactions and predict outcomes among variables. Descriptive statistics is the foundation for the entire system. It is used to define and explain the basic components in a study. Exploratory statistics tries to figure out what the collected data is saying. This method involves averages and percentages, which are usually displayed on a graph or in a table. Since by definition it relies on information from previous experiments, this data is sometimes called secondary research. Confirmatory statistics is the method that applies general ideas and concepts to an issue or a problem in an effort to answer specific questions.
7. Discuss the Probability Theory.
The Probability Theory is the study and analysis of random events, and especially whether those events can predict the behavior of a defined system. A probability is the numerical measure of the likelihood the event will happen. It is a number from zero (0) to one (1). Zero means it will definitely not happen, one means it definitely will happen and point five (0.5) means it is a draw, i.e. just as likely to happen as not happen. Probability is the possibility of an event happening or something being true. It is used to explain events that do not happen with any certainty. Probabilities must meet these general rules:
- All probabilities must be a non-negative number.
- The collection of all possible outcomes is equal to one (1).
- If there are two possible outcomes that cannot happen at the same time (non-overlapping), the possibility that either outcome will happen is the total of the individual outcomes.
8. Describe procedural and conceptual approaches to teaching math.
Teaching math with a procedural approach or by direct instruction means defining terms and symbols, explaining formulas, and giving students a step-by-step method to solve a problem. The teacher demonstrates the procedure and students practice the steps in class and try to work similar problems at home. Definitions are exact and necessary, but don't allow for creative examples or encourage critical thinking. The problem is solved "by the book." Most students acquire some level of proficiency but are usually unable to apply anything learned in math class to other academic areas or to situations outside of school. Teaching math with a conceptual approach means developing lessons and posing problems that require students to use their reasoning abilities, apply critical thinking skills, and make connections with previously-learned information. Knowing the definitions and formulas is necessary but is not the primary focus. The goal is to help students make sense of math by using examples to which they can relate, and making the lesson relevant to their life outside of math class.
9. Explain why it is better to focus on the conceptual learning approach and use the procedural approach as a teaching tool.
Experts agree that students need to know the definitions of terms, the applications of formulas, and the methods used to solve a problem. So even though studies have shown that using the procedural approach to teach math can actually inhibit understanding and prohibit integrating new concepts with previously learned data, the basics still need to be acquired during the study of the material. The question is when and how. Learning math necessarily entails improving reasoning ability, honing critical thinking skills, and discovering that these talents are applicable to other academic disciplines as well as to issues in the real world. To accomplish that goal, the teacher must design lesson plans, compose problems, and devise activities that require students to explain their thought process, compare methods and approaches, and justify results. With such instruction, students discover patterns and relationships and the activity becomes a meaningful learning experience rather than a rote exercise in memorization. Using this approach, students learn the definitions, formulas and methods as a natural outcome of understanding and integrating the new concepts.
10. Discuss the National Assessment of Educational Progress requirements for assessments in mathematics.
Mathematics involves the study of number sense, properties, operations, measurement, data analysis, statistics, probability, algebra and functions. The U.S. Department of Education has established criteria for testing comprehension of these math concepts using recommendations from the National Assessment of Educational Progress. Students are required to not only know facts and formulas but also to integrate the data into previously-acquired information by using critical thinking skills developed through studying these mathematical concepts. In other words, students need to be able to use the facts learned in math class in practical applications in the real world. The assessments developed by educators, curriculum specialists, and the business community emphasize the importance of assessing students' ability to reason, understand concepts, solve problems, evaluate results and communicate knowledge of the subject matter. The tests attempt to measure whether students can take cognitive skills learned in math class, apply them in other academic disciplines, and use them outside of school in meaningful ways.
Last Updated: 05/07/2014