The Labour Party has called for the resignation of the Minister for Education Dr Woods, over what it called his "incompetent" handling of the teachers' dispute.
The Labour Education spokeswoman, Ms Roisín Shortall, claimed the Minister had lost the confidence of parents and teachers and that the prospects of a swift end to the dispute would diminish even further if he stayed in office.
"The extraordinary decision of Minister Woods to travel to Malaysia and Singapore despite the huge crisis now facing second-level students is the last straw as far as many parents are concerned."
"The entire handling of this dispute by Minister Woods has, from the very beginning, been both incompetent and provocative," Ms Shorthall said.
"It is now also clear that the Minister's contingency plans for the holding of examinations without the participation of ASTI are collapsing around him, especially given the position now being adopted by the other teacher unions."
Secondary school teachers resumed industrial action today with a nationwide strike after the rejection of last week's Labour Court recommendation over their 30 per cent pay claim.
Almost 620 schools are closed as about 17,000 members of the Association of Secondary Teachers, Ireland take part in the first of six one-day strikes due to take place before Easter.
The ASTI has called for direct talks but as yet there has been no response to a letter sent to the Government on Monday, Ms Gemma Tuffy of ASTI said today.
The Taoiseach has said he will respond to the letter but has so far declined to negotiate directly with the ASTI.
A suspension on the strike action in January to facilitate the Labour Court ended over the weekend when ASTI's central executive committee voted overwhelmingly to reject the court's recommendation.
There is growing concern over the threat of disruption to the State's Junior and Leaving Cert exams.
The Department of Education's contingency plan to ensure the exams will go ahead has been opposed by ASTI and the Teachers' Union of Ireland.
Primary teachers and university lecturers have also said they would not participate in the contingency plans.
Teachers and opposition parties have also been critical of the Minister for Education Dr Woods's trip to Malaysia this week.
Dr Woods flew out last night and is representing the Government during St Patrick's Day celebrations.
Department spokesman Mr Brian Duggan defended the trip saying it had been planned for some time. He said: "Dr Woods is carrying out Government business in relation to the education sector as well as representing the Government abroad."
The Minister is to meet with his Malaysian counterpart, Mr YB Tan Sri Dato, and will sign a memorandum of understanding with Malaysia to provide for education co-operation between the two countries.
He is also to meet with Malaysian Prime Minister Mr Mahathir and attend the St Patrick Society Annual Ball in Kuala Lumpur.
Fine Gael's education spokesman, Mr Michael Creed TD, said last night the trip showed Dr Woods was not taking the dispute seriously enough.