Current models of number understanding highlight the role of the analogue magnitude system (AMS), a noisy system that enables both humans and animals to understand values. Additional systems extend the AMS to support exact number processing. In contrast, we propose an alternative framework for numerical cognition. (a) Number understanding is rooted in an object-based conceptual representation of natural numbers. (b) Discrete semantic system (DSS) is utilized to store and process exact values. Most of the numerical effects attributed to the AMS are actually caused by the DSS. (c) The meaning of the potentially abstract DSS originally can be found in the object-based conceptual understanding. Later the abstract layer can be anchored to additional domains (e.g., route, plane, AMS, etc.) to extend the meaning of numbers. This framework also suggests that preschoolers might understand exact numerosity before they understand specific values, and preschoolers might be able to understand counting earlier than they usually understand it. Several studies will be presented to support this alternative view.