We report on joint work with Aravind Asok investigating how having a rational point or zero cycle of degree one is reflected in the A^1-homotopy type of a complete variety. It turns out that from this point of view, the difference between having a rational point and admitting a zero cycle of degree one is a process of stabilization; this insight relies on a computation of the zeroeth stable homotopy sheaf of a variety in terms of Chow-Witt groups.