Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
Understanding of number develops in pre-kindergarten through grade 2 as children count and learn to recognize "how many" in sets of objects. A key idea is that a number can be decomposed and thought about in many ways (24 is 2 tens and 4 ones and also 2 sets of twelve). Beyond understanding whole numbers, young children can be encouraged to understand and represent commonly used fractions in context (1/2 of a cookie or 1/8 of a pizza). In grades 3 through 5 students should come to see fractions as parts of a unit whole or collection and as division of numbers. This focus on fraction concepts is central in upper elementary math instruction. In the grades 6-8 students should be able to move flexibly among equivalent fractions, decimals, and percents. By grades 9-12, other curricular areas are emphasized more than number, but these students should learn about the differences among number systems and their properties.

Understand meanings of operations and how they relate to one another.
In pre-kindergarten through grade 2, students should encounter a variety of meanings for addition and subtraction of whole numbers. Multiplication and division can begin to have meaning at this level using simple situations like how to share a bag of raisins fairly among four people. In grades 3-5, the central focus should be on helping students develop meaning for whole-number multiplication and division. Students should be able to choose an operation to solve a particular problem and should realize that the same operation can be applied in problem situations that might seem quite different from one another. In grades 6-8, the emphasis is on operations with rational numbers and on working with proportions. Students at this level also need to learn operations with integers. At the high school level, students will extend their understanding of operations, learning how to combine vectors and matrices arithmetically.

Compute fluently and make reasonable estimates.
By the end of grade 2, students should know the basic addition and subtraction facts, should be fluent in adding two-digit numbers, and should have methods for subtracting two-digit numbers. In grades 3-5, students should know the basic multiplication and division facts and should use reliable algorithms to solve arithmetic problems with whole numbers. Operations with fractions can be studied informally at this level, but grades 6-8 are a more appropriate level to develop computational fluency with rational numbers in fraction and decimal form. In the middle grades, students should also develop methods for computing with integers. At all levels students should make smart choices about when to use mental computation, paper-and-pencil strategies, estimation, and a calculator. A balance between conceptual understanding and computational proficiency needs to be struck from pre-kindergarten through 12th grade, for understanding without proficiency is as harmful as proficiency without understanding.