Understanding: Knowing what concepts, operations and symbols mean. Instead of just memorizing facts, students with understanding will know why a mathematical idea is important and in which context it is useful. They see patterns and see how numbers are related, such as connections between addition and subtraction and multiplication and division.
Computing: Carrying out mathematical procedures such as adding, subtracting, multiplying and dividing numbers, flexibly, accurately, efficiently and appropriately. It includes measurement (measuring items), algebra (solving equations), geometry (constructing figures) and statistics (graphing data). Being fluent means knowing not just how but when to use these procedures.
Applying: Formulating problems and devising strategies for solving them. A concept or procedure is not useful unless students know when and where to use it, and when and where it does not apply. In real life there are many situations when we are required to figure out what the problem is. Students need to be able to create problems, devise possible strategies, and choose the most useful strategy to solve the problem.
Reasoning: Using logic to explain or justify a solution or to extend from something known to something not yet known. One of the best ways for students to improve their reasoning is to explain or justify their solutions to others. As they communicate their thinking, they hone their reasoning skills.
Engaging: Seeing mathematics as sensible, useful, and doable, if you work at it, and being willing to do the work. There is no mysterious “math gene” that dictates success. Students should see mathematics not as an arbitrary set of rules and procedures, but instead a subject where things fit together logically and sensibly.