Is the Real Number Line Something to Be Built, or Occupied?

Klein emphasized geometry and intuition, and made the concept of function central to mathematics education. In fact, number and operations form the backbone of the school mathematics curriculum. A high school graduate should comfortably and capably meet an expression like, “Let y = f(x) be a function of a real variable x,” implying that the student has a robust sense of the real number continuum, the home of x. This understanding is a central objective of the school mathematics curriculum, taken as a whole. Yet there are reasons to doubt whether typical (U.S.) high school graduates fully achieve this understanding. Why? And what can be done about this? I argue that there are obstacles already at the very foundations of number in the first grades. The construction narrative of the number line, characteristic of the prevailing curriculum, starts with cardinal counting and whole numbers and then builds the real number line through successive enlargements of the number systems studied. An alternative, based on ideas advanced by V. Davydov, the occupation narrative, begins with pre-numerical ideas of quantity and measurement, from which the geometric (number) line, as the environment of linear measure, can be made present from the beginning, and wherein new numbers progressively take up residence. I will compare these two approaches, including their cognitive premises, and suggest some advantages of the occupation narrative.