To some mathematicians, mathematics is a beautiful subject which should be taught in a way which allows people to extend their minds and explore the universe. However, mathematics, and especially arithmetic, is also a necessity of modern life, and can be a struggle for many of us. Some teachers have advocated avoiding mental effort where possible to get the result quickly and painlessly. Mathematical instruments, devices and models have been invented with both aims in mind. This topic contrasts the two approaches – advocating mathematics and avoiding mathematics – from the eighteenth century until the present day, recognising that the tension goes back much further in the history of mathematical education.