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Some councillors have slammed the Prime Minister's suggestion schools could be opened in June.
Medway's Labour Group has expressed concerns at Tory Boris Johnson's plans for children in Reception, Year 1 and Year 6 to return next month.
Cllr Joanne Howcroft-Scott, who is part of the council's children and young people overview and scrutiny committee, said: "I know from my decades of experience as a senior leader, trade union representative and governor in schools the real concerns there are about this announcement.
"At a time when Covid-19 infection and death rates are still high, it is premature, to say the least, to reopen schools to those pupils who may be least able to maintain social distancing.
"My fear is that this move will put at risk our children, our teaching, leadership and support staff and they, along with parents and carers, have every right to be very worried about whether Boris Johnson’s move will lead to a second spike in infections."
Labour spokesman for the local authority's children services, Cllr Clive Johnson, said: "Medway has problems enough from the council’s poor stewardship of support for children, without the potential threat from a premature end to lockdown.
"Our schools need to be assured they will reopen only when it is safe to do so. We support the National Education Union’s call for its Five Questions to be answered before schools are allowed to open.
"We want to see much lower numbers of Covid-19 cases with a sustained downward trend maintained by effective contact tracing. There should be a clear national plan for social distancing and social mixing.
"This is particularly important for schools where school buildings and movement within them will make social distancing very difficult with large numbers of pupils attending.
"Schools need easy access to Covid 19 testing for pupils and staff to ensure prompt action if there are outbreaks and a clear strategy for what happens if an outbreak occurs, including the protection of vulnerable pupils, staff and members of the wider school community.
"Medway needs to be confident that there are sufficiently robust safeguards in place to protect our young people and those who look after them."
However, Medway Conservatives hit back at their opposition, saying their were "disappointed" by Labour members comments.
"Labour have deliberately misinterpreted or simply not understood clear government guidance..."
Councillor Josie Iles, portfolio holder for children's services and Martin Porter, portfolio holder for education and schools say they have been directly in touch with head teachers across Medway on a regular basis.
Cllr Iles said: “Schools have, throughout the unprecedented times of the last few weeks, continued to provide care for the children of key workers, and, from a council perspective, services such as our SEN school transport have also continued in order to provide support to parents.
"For this, the Conservative group are very thankful, and we sincerely appreciate the hard work of teachers and carers for their diligence and constant commitment to our young people.
"We know when schools fully reopen - and the Prime Minister has been clear that this will only be when it is safe to do so - they will do so in order to continue to put the needs of Medway’s children first.
"In contrast, Labour have deliberately misinterpreted or simply not understood clear government guidance for the sake of an opportunity to insult the hard work of those who have made sure that children across Medway have remained cared for and retained the stability they require.
"As we have come to expect from Labour, they also do not offer a single workable alternative model. Their recent comments betray a reluctance to give all of Medway’s children the chance to continue to receive the education they rightfully deserve through the phased reopening of schools.”
This week the government confirmed parents who don't send their children to children won't be fined.