To begin with, teachers must have a firm understanding of what mathematics truly is and what role it plays in their lives and the lives of their students. Though the two terms are often used interchangeably, mathematics is not arithmetic, but rather it's a way of thinking that involves reasoning, logic and problem-solving abilities. Mathematics studies patterns and relationships that are useful in all walks of life.
A qualified math teacher must be able to complete any and all math operations (such as addition, division, converting fractions and the like), and it's preferable that her knowledge extend to operations taught below and above her students' grade level. This ensures that she can not only educate her students in the procedures but can also give them an idea of where the concepts began and how this knowledge and practice will evolve.
Mathematics is full of descriptive words and vocabulary terms that need to be introduced, defined and committed to memory. A good teacher should never have to refer to her textbook to recall the term for the answer to a multiplication problem. She should know the terms well and should be able to explain them through various means to ensure the students comprehend.
Mathematics is not a mystery, but it's made up of vital formulas and equations that create the backdrop for problem-solving. These formulas, like the vocabulary, should be memorized, but beyond that, it's imperative the teacher understand and be able to explain the concept behind the formula or equation.
A teacher can be familiar with all the mathematical terms and operations, but unless she understands the concepts in such a way that she can apply them to everyday life, she will not succeed in fully educating her students. Mathematics is more than a list of rules and definitions, and a good teacher must help her students to understand that by showing them the many ways math can be used outside of the classroom.
Though mathematics is precise in that there can only be one answer to a particular problem, there are a number of ways by which an individual can arrive at that answer. A teacher must understand the problem and concept in such a way that she can demonstrate various ways to accomplish the task. This will also aid her in comprehending how students reach a certain solution when they deviate from the method she demonstrated.