In this article, which was first published in 1991, the late Sandy Dawson, discusses aspects of a Lakatosian approach to mathematics teaching. The ideas are illustrated with examples from three teaching situations: making conjectures about the next number in a sequence; making conjectures about the internal angles in a triangle using Logo; and developing a way to find the difference between two eighteen-digit numbers entirely as a mental calculation. The article appeared as a chapter in Teaching and Learning School Mathematics, edited by David Pimm and Eric Love, co-published by the Open University and Hodder and Stoughton, and reprinted with permission.