Dozens of local authorities across the country have advised schools not to participate in wider reopenings today, new research shows.
Some councils are warning that the test and trace programme is not yet "robust enough" to sufficiently reduce Covid-19 transmission in schools, where social distancing is hard to maintain with children, according to the PA news agency.
A survey of councils carried out by PA suggests the number of pupils back in the classroom today varied significantly, depending on the local area.
The survey showed that dozens of local authorities across England, predominantly in the North, advised against a return to school on 1 June amid safety concerns.
Among them is Hartlepool Borough Council, which told Tes that neither maintained schools nor academies in the area had engaged in wider reopenings today.
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But in some areas of England – such as Kingston and Richmond in London – local authorities reported that the vast majority of primary schools were providing some provision for priority year groups from today.
Hertfordshire County Council said this morning that more than 90 per cent of the county's primary schools had reopened to more pupils.
Schools, colleges and nurseries closed 10 weeks ago due to the Covid-19 outbreak, remaining open only for vulnerable youngsters and the children of key workers.
It is not yet clear how many parents opted to send their children back to school today but a recent survey suggests that heads were expecting around half of families to keep pupils at home.
Cathy Moden, headteacher of Hiltingbury Infant School in Chandler's Ford, Hampshire, said she had anticipated 45 of the 90 children in Reception to attend on Monday but only 39 turned up.
Ms Moden said: "I do expect it to increase, I think some parents have made a decision on what they have heard in the media. I have heard from some parents they aren't ready yet to send their children."
Kieron Smith, from Blyth in Northumberland, who has a son in Reception, said he will not be sending his child back to school until he has more confidence in the government's approach.
"All in all, it's not worth it. It's not a risk I'm able to tolerate. The government have not assured us of our children's safety," he said.
The government's aim is for all primary school pupils to return to school before the summer break.
In a message to headteachers ahead of the reopening, Paul Whiteman, general secretary school leaders' union the NAHT, said: "We will be arguing that a full return before the summer break is not possible based on the advice we have from the Department for Education.
"Despite the narrative from the government, the level of confidence for a return to schools remains low.
"The next few days will reveal whether the government has passed the confidence test. We will discover how many families feel confident to come to school. And we will get a sense of the reaction from staff too."
Business secretary Alok Sharma said he understood parents' concerns about sending their children back to school.
He told BBC One's Breakfast programme: "I completely understand every parent wants to keep their child safe. And that's precisely what the government wants to do in ensuring schools are safe places to return to."