David Hart, who stepped down as general secretary of the National Association of Head Teachers last month, makes the claim in a paper submitted to the Government.
He said it was strange that nine and 10-year-olds were not taught in classes by subject when it was common practice in private schools and middle schools.
"I have always wondered why primary teachers are treated as jacks of all trades, teaching all the core and most of the foundation subjects to their classes," he writes.
"The fact this has existed since time immemorial is no excuse. It is based on a rather quaint and snobbish, yet highly damaging, belief that 10-year-olds need teachers that are all-rounders but 11-year-olds require specialists."
Mr Hart said that unless urgent action was taken to improve the specialist knowledge of teachers responsible for the last two years of primary education, the country would not stand "a cat in hell's chance" of raising standards to the levels demanded by global competition.
He proposed that primary schools should consider rewriting their timetables so that teachers would take lessons by subjects, rather than classes, at least for the final two years of pupils' primary education.
Mr Hart suggested schools should also make better use of support staff in areas such as sport, languages and performing arts where some primary teachers would have gaps in their knowledge.
Primary teachers, especially those teaching the top two years, would need greater training to become subject specialists. But Mr Hart said primary schools could also benefit from the redundancies that falling rolls in secondary schools will cause in the near future.
He was invited to submit the paper by the Department for Education and Skills after conversations with officials while he was still NAHT general secretary.
His roles now include developing ideas for a leadership programme for academy headteachers, which is being overseen by the Specialist Schools and Academies Trust.
Mick Brookes, Mr Hart's successor as NAHT general secretary, said primary schools were keen to broaden the curriculum but would oppose greater specialisation by subject. Mr Brookes, who is a former primary headteacher, said: "We wouldn't want to go back to seeing secondary-style lessons in primary schools. The move in primary schools has been in the opposite direction, towards making the curriculum more joined up."
Robina Crowe, head of Smithdown primary in Liverpool, also believes that a secondary school-style timetable would not work with her pupils.
"We have a lot of EAL (English as an additional language) children who need the continuity and support that a single class teacher can provide," she said.
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