Thomas Kuhn (1962/2012) introduced the term "paradigm shift" to the scientific literature to describe how knowledge in science develops. The aims of this article are to identify paradigm shifts, or revolutions, that have occurred in mathematics, and to discuss their relevance to teaching mathematics in schools. The authors argue that Kuhn's ideas could be used to promote awareness that mathematics adapts to different cultural and historical contexts. In this article, they give a brief outline of the key ideas in Kuhn (1962/2012). Then, they consider its relevance to the development of mathematics. Have there been revolutions in mathematics? They go on to explore links between Kuhn's ideas and Piaget's theory of constructivism. They propose five topics from the "Australian Curriculum: Mathematics" that could be regarded as revolutionary in the sense of Kuhn: the Hindu-Arabic numeral system, algebra, irrational numbers, coordinate geometry, and infinity. They then suggest a sequence of lessons aimed at presenting one of these topics to middle years and senior secondary classes. Conclusions are presented in the final section.