Type I Knowledge: This mathematics knowledge is rotely learned and employs memorization. It includes memorized knowledge of definitions, procedures, or rules. Teachers with this knowledge can rotely perform skills, apply rules, and give definitions.
Type II Knowledge: This mathematics knowledge is conceptual in nature. It includes a deep understanding of mathematical concepts, procedures, laws, principles, and rules. It is knowledge of connections and relationships among concepts. It is often associated with meaning. Teachers with this knowledge can give examples/non-examples and identify properties/ characteristics of mathematical concepts. They can compare and contrast and represent mathematical concepts and generalizations in multiple ways. They can explain and create mathematical procedures and represent them in multiple ways.
Type III Knowledge: This mathematics knowledge is higher order in nature. It includes applying knowledge to solve problems and real-world applications. Teacher with this knowledge can reason informally and formally, conjecture, validate, analyze, and justify. They can use deductive, inductive, proportional, and spatial reasoning to solve problems.
Type IV Knowledge: This mathematics knowledge is unique to teaching mathematics. It represents the mathematics knowledge that teachers use in the act of teaching. It includes knowledge of the most regularly taught topics in mathematics, the most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations. Teachers with this knowledge can identify student misconceptions about mathematics and provide strategies to correct them. Teachers can derive activities that promote understanding, reasoning, and proficiency. They can provide examples, analogies, models, or representations to help students understand mathematical concepts or procedures.