In the early stages of learning a concept, children are presented with physical objects such as figurines, wooden or plastic shapes, base ten blocks, counters, and number lines to help form a concrete understanding of the concept.

As learning of a concept progresses, the concept is then represented pictorially in the form of drawing direct representations of objects in a problem or by using more abstract representations like number lines, base ten block drawings, and bar models. This helps transition students away from relying on physical objects to understand a concept, but still provides visuals that can help solidify understanding.

In the later stages of learning a concept, abstract forms of representations–including numbers, symbols, and algorithms–are introduced to promote efficient problem-solving and a more mature understanding of the mathematical concept.

These approaches are used in all grade levels, and are often presented together, but as we move from younger years into later years, representations move from mostly concrete to mostly pictorial and abstract.

The sequence of topics is carefully chosen. Topics introduced early on are revisited at more sophisticated levels as students progress in the program, which helps reinforce and expand the understanding of mathematical concepts and skills. The C-P-A approach mentioned above is one of the methods that helps students strengthen their understanding as complexity grows.

Relational understanding refers to the ability to explain a procedure. Instrumental understanding refers to the ability to perform a procedure (e.g., using an algorithm, mental math strategy, or formula to calculate a solution). Singapore Math relies on both of these principals, but the main goal is to ensure students can correctly explain how they solved a problem (relational understanding), which will typically involve using procedures (instrumental understanding).

Problems are presented in various ways throughout a lesson, and students may be asked to show their solutions using different representations (e.g., with base ten blocks, a drawing, and an algorithm). This systematic variation helps students deepen their conceptual understanding; recognize how procedures, skills, and concepts can be applied in multiple ways; and equips students with more than one strategy to solve a problem.