Symbolic mathematics allows humans to represent and describe the logic of the world around us. Although we typically think about math symbolically, humans across the lifespan and a wide variety of animal species spontaneously exhibit numerical competence without reference to formal mathematics. This intuitive ability to approximately compare, estimate, and manipulate large non-symbolic numerical quantities without language or symbols is called the Approximate Number System. The four chapters of this dissertation explore whether non-symbolic, approximate calculation can function as a bridge between our Approximate Number System and symbolic mathematics for children at the beginning of formal math education and university undergraduates. Chapter 1 explores how non-symbolic and symbolic ratio reasoning relates to general math skill and Approximate Number System acuity in elementary school children. Chapter 2 examines whether children and adults can perform a non-symbolic, approximate division computation, and how this ability relates to non-symbolic and symbolic mathematical skill. Chapter 3 tests the robustness and mechanism of a non-symbolic, approximate addition and subtraction training paradigm designed to improve arithmetic fluency in university undergraduates. Chapter 4 investigates whether the negative relation between math anxiety and symbolic math performance extends to approximate, non-symbolic calculation. Together, Chapters 1 and 2 provide evidence that non-symbolic calculation ability functions as a mechanism of the relation between Approximate Number System acuity and symbolic math. Chapters 3 and 4 identify populations of students for whom practice with non-symbolic calculation may or may not be beneficial. In sum, this dissertation describes how non-symbolic, approximate calculation allows students harness their intuitive sense of number in a mathematical context.