Professor Wragg suffered a cardiac arrest on one of his habitual early-morning runs. He was taken to hospital but died early on Thursday morning. He was 67.
Since September 1980, his columns have appeared regularly in The TES, satirising government bureaucracy and jargon, and attacking league tables and the culture of testing. And he laid into the Ofsted inspection regime mercilessly, saving his most vicious criticism for Chris Woodhead, the former chief inspector.
He also wrote satirical sketches for Rory Bremner, the impressionist, with whom he said he shared a common interest in "sending up the ludicrous".
Professor Wragg coupled this role as scourge of the establishment with friendships across a range of politicians and education officials.
As Exeter university's professor of education between 1978 and 2003, he advised the education secretaries Estelle Morris and David Blunkett, and regularly discussed schools with John Prescott, the deputy prime minister, who paid tribute to his "enthusiasm, his vision and his great humour".
Since 1974, Professor Wragg has written more than 40 books, which have been translated into a range of languages. He also published more than 800 articles in books, research journals, newspapers and magazines.
He held a range of positions within education, including member of the board of the Qualifications and Curriculum Authority, and chairman of judges for the national teaching awards. David Puttnam, chair of the teaching awards trust, said: "He understood that the whole thing was about the individual teacher."
Indeed, most teachers felt they were his first priority. He continued to work in schools, maintaining a particularly close link with those in Exeter. Geoff Williams, headteacher of St Leonard's primary, in Exeter, said: "He spoke using language that was understandable to everybody, and not filled with education jargon.
"He was never the remote leader. He responded to people's individual expressions. He was universally admired, and a man of great compassion."
Classroom tributes have poured into the TES website. One contributor, Michael Tidd, wrote: "There are few people who can claim to be so consistently popular with a group so hard to please as teachers. Ted Wragg managed that because he spoke the language of common sense in education."
After his retirement in 2003, he continued to teach, write and speak as an emeritus professor. Most recently, he led an experiment in transforming tearaway teenagers, screened as The Unteachables on Channel 4.
David Hargreaves, former chief executive of the QCA, said: "Many TES readers would go home at the weekend, read Ted's column and burst out laughing. Life seemed much more bearable when Ted was making fun of the things that exasperated them. Many a weekend won't be the same now."
Ted Wragg is survived by Judith, his wife, Maria, his mother, three children and three grandchildren. A private family funeral service is being held today.
Memorial services in Exeter and London are also planned.
See tributes at www.tes.co.uk
ObituAry and extracts from Ted's columns, 16-17