Proficiency in mathematics means that students can solve meaningful, challenging problems; demonstrate both a depth and breadth of mathematical understanding; and perform both simple and complex computations and mathematical procedures quickly and accurately, with and without the aid of computational tools. Many mathematicians, educators, and experts on children's mathematical thinking and learning view these components as ideally interdependent.

Three Ways of Knowing

Mathematics education must provide students with a balanced instructional program. In such a program students a) become proficient in basic computational and procedural skills, b) develop conceptual understanding, and c) become adept at problem solving and mathematical reasoning. All three components are important; none is to be neglected or underemphasized.

• Computational/Procedural Skills: Mathematics skills that all students should learn to use routinely and automatically, such as basic addition, subtraction, and multiplication tables in elementary to more complex number facts in middle and high school. Basic computational and procedural skills develop over time, and they increase in depth and complexity through the years.
• Conceptual Understanding: Conceptual understanding is the knowledge of the underlying concepts, structures and logic of mathematics. It includes knowing not only how to apply skills but also when to apply them and why they are being applied. Seeing the structure and logic of mathematics and using it flexibly, effectively, and appropriately.
• Problem Solving: Problem solving in mathematics is a goal-related activity that involves applying skills, understandings, and experiences to resolve new, challenging, or perplexing mathematical situations. Problem solving involves a sequence of activities directed toward a specific mathematical goal, such as solving a word problem, a task that often involves the use of a series of mathematical procedures and a conceptual representation of the problem to be solved.

- From CA Mathematics Framework, 2000