This presentation proposes a re-framing of the way educators see and teach numeracy in primary, middle and high school mathematics courses. Traditionally, numbers are first introduced as purely quantitative entities used for measurement and record-keeping, often leading to confusion as more and more abstract properties are eventually implied via the familiar arithmetic operations. By contrast, a group-theoretical approach to number sense would begin by constructing an object that first has the required properties, and then showing that it must be a number. Beginning with “logic games” in primary grades, and progressing through the construction of the familiar number sets and their corresponding operations from first principals in the middle grades and above, this presentation aims to provide suggestions and examples of how to integrate more abstraction into the traditional math curriculum.