Headteachers are to be given the power to dismiss teachers who are members of the BNP or other groups which have an "extremist tenor".

The pledge by the education secretary, Michael Gove, follows the case of a BNP activist who used a school laptop to post comments describing some immigrants as "filth". Gove said he would allow school heads and governing bodies to sack teachers for membership of the far-right party. Members of the BNP are barred from working as police or prison officers.

The minister told the Guardian: "I don't believe that membership of the BNP is compatible with being a teacher. One of the things I plan to do is to allow headteachers and governing bodies the powers and confidence to be able to dismiss teachers engaging in extremist activity.

"I would extend that to membership of other groups which have an extremist tenor. I cannot see how membership of the British National party can co-exist with shaping young minds."

The crackdown was welcomed by the NASUWT teaching union, which has called for a ban. Its general secretary, Chris Keates, said: "I hope this is something Michael Gove takes forward as quickly as possible. It is an important part of safeguarding the interests of young people."

Adam Walker, a BNP member and teacher at a school near Sunderland, was cleared of racial and religious intolerance by a disciplinary panel in June this year. He wrote in an online forum that Britain was a "dumping ground for the filth of the third world". But while the panel said the comments were troubling, it was not satisfied that they showed intolerance.

Delivering the General Teaching Council's verdict, the chair, Angela Stones, said: "A negative comment about immigration to the UK of itself need not be indicative of racist views or racial intolerance, since the race of immigrants is extremely varied."

The GTC, the profession's watchdog, was abolished a week later. In a Commons statement, Gove cited the decision made on the Walker case as "quite wrong".

The previous government ruled out banning BNP members from teaching after an independent inquiry into racism in schools decided it would be disproportionate. Maurice Smith, a former chief inspector of schools, said a ban would be "taking a very large sledgehammer to crack a minuscule nut". He also described it as a "profound political act".

The review found that in the past seven years four teachers and two governors had been publicly identified as members of racist organisations, and that only nine incidents of teachers making racist remarks or possessing racist material had been subject to GTC disciplinary sanction.

Critics say this review over-emphasised the number of incidents, rather than their potential impact on a classroom. Keates said: "It's not an issue of scale. That was our biggest criticism of Maurice Smith's report. T the issue is one of making sure that a clear message goes out that membership of the BNP and other extremist groups is incompatible with being a teacher."

Gove outlined plans to ban the BNP from classrooms in an interview in which he also said he would not oppose parents and teachers if they rejected his traditionalist views on the curriculum.

At this year's Conservative party conference, Gove said

historian Simon Schama would advise on an overhaul of the history curriculum to ensure that no pupil leaves school without learning "narrative British history". English teaching will also be reformed to ensure that canonical writers such as Pope, Shelley, Dryden, Dickens and Hardy are at the heart of the curriculum.

He said he believed there was an "unfulfilled appetite" for a more classical education. But he insisted teachers could steer their own course. "The idea of the reforms I'm about to bring about is that if the professional wisdom of teachers takes them in a different direction, and the common sense of parents takes that direction, I won't stand in their way."