The draft changes, announced this week by the Qualifications and Curriculum Authority, aim to give schools more freedom over what they teach and make it easier to tailor subject content to pupils' needs. But they also prescribe some changes. Maths will have to include the subject's history and its applications, such as digital technology.
In PE, swimming has been dropped from the six options available to teachers, in favour of "fitness and health activities".
David Sparkes, chief executive of British Swimming, said: "It is important to the nation that our young children learn to swim and have a knowledge of water safety. Swimming is a life skill and prepares children for an active life."
For the first time, the history curriculum specifically mentions the "British empire". At present, pupils study, under the heading "Britain 1750-1900", "How expansion of trade and colonisation, industrialisation and political changes affected the United Kingdom". The new draft says study will include: "The British empire and its impact on different people."
Jerome Freeman, the exam watchdog's history adviser, said: "Empire has been given more emphasis and greater strength. It is a significant part of British history and a topic that history teachers have to embrace more."
Mr Freeman said the aim had been to keep the balance between world and British history. But the dropping of two compulsory world history topics and one on European history has sparked concern. Nathan Cole, head of history at Saffron Walden high - who was asked to comment on the curriculum - said: "It is very British-centric. It does not mention world and European history at all."
There are also changes to English. The existing curriculum says 11-to-14-year-olds should study a Shakespeare play, eight major poets and four major fiction writers, half before and half after 1914, and drama by major playwrights. Teachers must choose from a prescribed list of pre-1914 authors and poets, including Austen and Wordsworth. This list has now been scrapped, though the new draft still requires "at least one Shakespeare play".
Bethan Marshall, lecturer in English education at King's College, London, said English teachers would welcome the news. She said: "Anything that cuts what teachers have to do is a good thing because you have to have professional trust."