The two men were appointed joint vice-chairs of the task force by David Blunkett, the Education Secretary, in a move interpreted as an attempt to unite opposing factions. However, relations between Mr Woodhead and Professor Brighouse deteriorated.
In a private letter to Professor Brighouse in February, the chief inspector accuses him of having "taken every opportunity to undermine the Office for Standards in Education" and says that such behaviour makes difficult his (Woodhead's) continued participation in the standards task force.
It would appear from the letter, leaked to The TES, that Mr Blunkett had already appealed to both men not to attack each other in public. The event that had particularly angered Mr Woodhead was the appearance of Professor Brighouse before MPs on the Commons select committee.
The letter says: "If you really think that your comments to the select committee were 'moderate and constructively critical', then you are either speaking a different language to the rest of us or you are deluding yourself... You have taken every opportunity you can to undermine OFSTED. It disappoints me, given your promise to David, and it makes my continued participation in the standards task force difficult."
In the event, Professor Brighouse resigned from the task force last month, saying he wanted to devote his energies to raising standards in Birmingham where he is education director. The following week, Mr Blunkett asked him to become special adviser to ministers on inner cities.