Students move through stages of understanding about addition that demonstrate cognitive shifts in mathematical thinking. Developmentally, they begin additive tasks by having to physically touch and count visible items. Next, they are able to re-present items, using verbal, motor, or figural representations that are not visible. Later they, develop an understanding of using the initial number as a composite—the first number is a known quantity. For example, a student is able to solve 9 + 3 by saying “nine…ten, eleven, twelve” without have to re-present the nine items visually or verbally.