Geoff Barton of the head teachers' union ASCL said he was disappointed that outgoing Cabinet member Justine Greening, a former comprehensive school pupil, had been replaced by Mr Hinds, who was educated at a Catholic grammar school in Altrincham before heading to Oxford's Trinity College where he studied politics philosophy and economics.
A public row has broken out between a ministerial ally of Justine Greening and Theresa May's former chief of staff after the former education secretary was accused of blocking plans to reform tuition fees.
"Hinds must be courageous enough to do that, to ensure universities are better, fees are lower and young people get the technical or academic education that suits them".
Writing in the Telegraph, Mr Timothy said he welcomed Ms Greening's removal from cabinet.
Mr Johnson said Ms Greening had been "faultlessly loyal".More news: Brexit caused 37% fall in new London financial jobs in December
"Charged with making Britain "the world's great meritocracy", she put the brakes on policies that work, like free schools, and devised bureaucratic initiatives of little value".
But in her role as education secretary, Greening was particularly at odds with Catholic bishops in the United Kingdom over her personal objection to removing the cap on new faith schools' admissions.
Mr Timothy, who quit Downing Street after playing a key role in the Conservatives' unsuccessful general election campaign, has since become a newspaper columnist.
The claim has been refuted by the Tory.
Justine Greening was replaced as Secretary of State for Education during Theresa May's Cabinet reshuffle.More news: Donald Trump says he would 'beat Oprah' in presidency battle
"This is obviously completely untrue, and I played no part whatsoever in the reshuffle".
Mr Barton said: "She has tried hard to tackle the school funding crisis, without any help from the Chancellor or Prime Minister".
He denied lobbying for Ms Greening's sacking but described Mr Hinds" promotion as "the bright point in an otherwise limited set of changes'.
David Davis, Boris Johnson, Phillip Hammond and Amber Rudd all retained their jobs, as did Jeremy Hunt and Greg Clark after the pair refused to move Department.
Greening was sacked as Education Secretary this week and left the government after refusing a new post of Work and Pensions Secretary.More news: Tonya Harding's Longtime Rep Reportedly Quits Over Ridiculous Demands!