Columnar joints in basaltic lava flows display conspicuous bands oriented normal to column axes. New observations show that each band contains a single plumose structure and thus represents an individual crack, or joint segment, formed during a discrete growth event. Analysis of plumose structure and intersections of cracks leads to a new kinematic model of columnar jointing, and provides the first direct proof that columnar joints grow incrementally from exterior to interior regions of solidifying magma bodies. Columnar joints form by nucleation and growth of new cracks on the edges of older cracks. Each new crack begins at a point and propagates mostly normal to column axes and along the leading edge of a developing column face, where thermal stress is concentrated. Inward propagation of cracks toward hotter regions is limited by a decrease of thermal stress and by the brittle-ductile transition of lava; outward and lateral propagation is limited by mechanical interaction with previous cracks and by low thermal stress in already fractured lava. Cracks often diverge slightly from the planes of previous cracks, probably because of spatial and temporal changes in directions of local principal stresses. Mechanical interaction causes a diverging crack to overlap, curve toward, and usually intersect the previous crack behind its edge, leaving a blind tip that points in the overall growth direction of the columnar joints. This and other directional criteria are applied to determine joint-growth patterns in several lava flows of the western United States. In two-tiered and multi-tiered flows, downward-growing columnar joints usually meet upward-growing joints well below the middle of the flows, which indicates very rapid cooling of upper portions relative to lower portions. This supports the idea that convection of water in columnar joints connected to the surface may be an important mechanism for cooling the upper portions of these flows, whereas conduction is probably the dominant cooling mechanism at the bases.