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All other GCSE coursework would have to be supervised, Mr Johnson told the Labour party conference on Wednesday. Wales is to follow the same course.
Mr Johnson said qualifications could not be undermined by a few cheats who took their work from the internet.
Parents needed to be reassured that coursework assessed pupils' work in a "fair and robust way".
Coursework - or work done outside normal classroom conditions - was introduced when GCSEs replaced O-levels almost 20 years ago.
I hope that coursework is retained for subjects such as the arts, geography and history
But there has long been a problem over its use, with concerns about whether work done outside school is always entirely the student's own efforts.
And the growth of model answers on the internet has increased concerns.
In many subjects, coursework is typically worth between 25% and 40% of marks.
Coursework is generally popular with teachers, who argue it allows pupils to show a wider range of skills and knowledge than traditional, timed exams.
'Robust and reliable'
In his speech to the Labour Party conference, Mr Johnson said: "Last year, we asked the Qualifications and Curriculum Authority to look at making GCSE coursework more robust and reliable.
"As a result of the QCA's report, we will be removing all GCSE coursework from maths and stipulating that in other subjects, coursework must be supervised in classroom style conditions.
Coursework is typically worth between 25% and 40% of marks
"Coursework can be very beneficial and I am determined that the hard work of the vast majority of students should not be undermined by questions of validity.
"We will work closely with teachers to develop even more effective and reliable coursework assessments."
In Wales, Education Minister Jane Davidson said she shared his concern.
"I shall ask my officials to work closely with teachers in Wales to ensure that these developments are taken forward in ways that are manageable for teachers and their pupils."
Shadow education secretary David Willetts said Mr Johnson needed to go further in cracking down on the use of coursework in other subjects where it was open to abuse.
Liberal Democrat education spokeswoman, Sarah Teather, said: "We have to be careful not to disadvantage those students who don't perform well in the exam hall environment.
"We mustn't simply have a knee-jerk reaction against coursework."
National Union of Teachers general secretary Steve Sinnott said: "We welcome the removal of unnecessary coursework.
"There are some subjects for which it is totally inappropriate. I hope however that coursework is retained for subjects such as the arts, geography and history."
The NASUWT teachers' union said it would examine the proposals very carefully but was sure supervised coursework would address problems of plagiarism.
The Association of School and College Leaders praised the announcement.
"In the age of the internet, plagiarism represents a real problem and it is right that the regulations on carrying out coursework recognise this," said general secretary Dr John Dunford.
Last week, it was announced that the coursework component in geography A-level would be scrapped, although in English it could be a greater amount than now, and it has been made compulsory for the first time in history.
Mr Johnson also outlined new measures to improve support for children in care and their education.
He said an extra £100 a year would be put into their Child Trust Fund accounts for every full year they spent in care.
And a £2,000 bursary would help children in care go to university.