Students with number sense naturally decompose numbers, use particular numbers as referents, solve problems using the relationships among operations and knowledge about the base-ten system, estimate a reasonable result for a problem, and have a disposition to make sense of numbers, problems, and results. For example, children in the lower elementary grades can learn that numbers can be decomposed and thought about in many different ways–that 24 is 2 tens and 4 ones and also two sets of 12.
Computational fluency–having and using efficient and accurate methods for computing–is essential. Students should be able to
perform computations in different ways, including mental calculations, estimation, and paper-and-pencil calculations using mathematically sound algorithms. All students should use calculators at appropriate times, setting the calculator aside when the instructional focus is on developing computational algorithms. Computational fluency should develop in tandem with understanding.
Problem solving is an integral part of all mathematics learning. In everyday life and in the workplace, being able to solve problems can lead to great advantages. However, solving problems is not only a goal of learning mathematics but also a major means of doing so. Problem solving should not be an isolated part of the curriculum but should involve all Content Standards.
Problem solving means engaging in a task for which the solution is not known in advance. Good problem solvers have a “mathematical disposition”–they analyze situations carefully in mathematical terms and naturally come to pose problems based on situations they see.
Students need to develop a range of strategies for solving problems, such as using diagrams, looking for patterns, or trying special values or cases. These strategies need instructional attention if students are to learn them. However, exposure to problem-solving strategies should be embedded across the curriculum. Students also need to learn to monitor and adjust the strategies they are using as they solve a problem.